Theorem uniexd 4537

Description: Deduction version of the ZF Axiom of Union in class notation. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypothesis

Ref	Expression
uniexd.1	|- ( ph -> A e. V )

Assertion

Ref	Expression
uniexd	|- ( ph -> U. A e. _V )

Proof of Theorem uniexd

Step	Hyp	Ref	Expression
1		uniexd.1	. 2 |- ( ph -> A e. V )
2		uniexg 4536	. 2 |- ( A e. V -> U. A e. _V )
3	1, 2	syl 14	1 |- ( ph -> U. A e. _V )

Syntax hints:   -> wi 4   e. wcel 2202   _Vcvv 2802   U.cuni 3893

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-uni 3894

This theorem is referenced by:  supex2g  7231  ptex  13346  zrhval  14630  zrhvalg  14631  zrhex  14634