Confusion about gödels proof for con(ZF+CH)

Hey I'm trying figure out how Gödel proved the relative consistency of
CH. As far as I understand his proof was like:

1. Assume that ZF is consistent. Then it has a model which we call V.

2. Construct L by choosing only those sets from V that is definable
by a first-order formula.

3. Show that L is a model of ZF+CH and thus ZF+CH must be
consistent.

My problem is the first part. How did he know that consistency
implied the existence of a model? I thought that this was not proven
until 1947? I have seen some people refer to this result as "Gödel's
completeness theorem", but I believe that this is not accurate?

Also, isn't V more that just a model? Doesn't it have properties that
is not implied by ZF like for example that ? is interpred as real
membership?
.

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