Theorem bj-uniex 16926

Description: uniex 4581 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 3942	. . 3 |- ( x = A -> U. x = U. A )
3	2	eleq1d 2307	. 2 |- ( x = A -> ( U. x e. _V <-> U. A e. _V ) )
4		bj-uniex2 16925	. . 3 |- E. y y = U. x
5	4	issetri 2831	. 2 |- U. x e. _V
6	1, 3, 5	vtocl 2877	1 |- U. A e. _V

Syntax hints:   = wceq 1402   e. wcel 2209   _Vcvv 2821   U.cuni 3933

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-14 2212  ax-ext 2220  ax-un 4576  ax-bd0 16822  ax-bdex 16828  ax-bdel 16830  ax-bdsep 16893

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-rex 2534  df-v 2823  df-uni 3934

This theorem is referenced by:  bj-uniexg  16927  bj-unex  16928