Theorem bj-uniex 16674

Description: uniex 4557 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-uniex.1	|- A e. _V

Assertion

Ref	Expression
bj-uniex	|- U. A e. _V

Proof of Theorem bj-uniex

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-uniex.1	. 2 |- A e. _V
2		unieq 3922	. . 3 |- ( x = A -> U. x = U. A )
3	2	eleq1d 2301	. 2 |- ( x = A -> ( U. x e. _V <-> U. A e. _V ) )
4		bj-uniex2 16673	. . 3 |- E. y y = U. x
5	4	issetri 2822	. 2 |- U. x e. _V
6	1, 3, 5	vtocl 2868	1 |- U. A e. _V

Syntax hints:   = wceq 1398   e. wcel 2203   _Vcvv 2812   U.cuni 3913

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-un 4553  ax-bd0 16570  ax-bdex 16576  ax-bdel 16578  ax-bdsb 16579  ax-bdsep 16641

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-v 2814  df-uni 3914  df-bdc 16598

This theorem is referenced by:  bj-uniexg  16675  bj-unex  16676