CSS Animation and Display None

Question

I have a CSS Animation for a div that slides in after a set amount of time. What I would like is for a few divs to fill the space of the animated div that slides in, which it will then push those elements down the page.

When I attempt this at first div that slides in still takes up space even when it is not visible. If I change the div to display:none the div doesn't slide in at all.

How do I have a div not take up space until it is timed to come in (using CSS for the timing.)

I am using Animate.css for the animations.

Here is what the code looks like:

<div id="main-div" class="animated fadeInDownBig"><!-- Content --></div>

<div id="div1"><!-- Content --></div>
<div id="div2"><!-- Content --></div>
<div id="div3"><!-- Content --></div>

As the code shows I would like the main div to be hidden and the other divs show at first. Then I have the following delay set:

#main-div{
   -moz-animation-delay: 3.5s;
   -webkit-animation-delay: 3.5s;
   -o-animation-delay: 3.5s;
    animation-delay: 3.5s;
}

It is at that point that I would like the main div to push the other divs down as it comes in.

How do I do this?

Note: I have considered using jQuery to do this, however I prefer using strictly CSS as it is smoother and the timing is a bit better controlled.

EDIT

I have attempted what Duopixel suggested but either I mis-understood and am not doing this correctly or it doesn't work. Here is the code:

HTML

<div id="main-div" class="animated fadeInDownBig"><!-- Content --></div>

CSS

#main-image{
    height: 0;
    overflow: hidden;
   -moz-animation-delay: 3.5s;
   -webkit-animation-delay: 3.5s;
   -o-animation-delay: 3.5s;
    animation-delay: 3.5s;
}
#main-image.fadeInDownBig{
    height: 375px;
}

Answer 1

CSS (or jQuery, for that matter) can't animate between display: none; and display: block;. Worse yet: it can't animate between height: 0 and height: auto. So you need to hard code the height (if you can't hard code the values then you need to use javascript, but this is an entirely different question);

#main-image{
    height: 0;
    overflow: hidden;
    background: red;
   -prefix-animation: slide 1s ease 3.5s forwards;
}

@-prefix-keyframes slide {
  from {height: 0;}
  to {height: 300px;}
}

You mention that you're using Animate.css, which I'm not familiar with, so this is a vanilla CSS.

You can see a demo here: http://jsfiddle.net/duopixel/qD5XX/