Theorem sepab 4273

Description: Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.) Put in closed form. (Revised by BJ, 18-Jul-2026.)

Assertion

Ref	Expression
sepab	|- ( A e. V -> { x | ( x e. A /\ ph ) } e. _V )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   V( x)

Proof of Theorem sepab

Step	Hyp	Ref	Expression
1		id 19	. 2 |- ( A e. V -> A e. V )
2		ssab2 3332	. . 3 |- { x | ( x e. A /\ ph ) } C_ A
3	2	a1i 9	. 2 |- ( A e. V -> { x | ( x e. A /\ ph ) } C_ A )
4	1, 3	ssexd 4268	1 |- ( A e. V -> { x | ( x e. A /\ ph ) } e. _V )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2209   {cab 2224   _Vcvv 2821   C_ wss 3220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-sep 4244

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233

This theorem is referenced by: (None)