That is true of fires you normally see. Fire is an example of an oxidation reaction but oxigen is not the only oxidizing agent. In a redox reaction the oxidizing agent is any species that can accept electrons - and the reducing agent is one that donates electrons (confusing terminology isn't it). So element such as chlorine and fluorine can also be oxidizing agents.
Also the oxidizing agent doesn't have to be present as a gas. In APCP the ammonium perchlorate (NH4ClO4) is the oxygen donor and some metal (typically aluminium) is the reducing agent. APCP has the advantage that while donating oxygen it decomposes to yield nitrogen gas (amongst other things) thus increasing the volume of the material dramatically and so adding to the thrust.
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