Theorem bj-uniex 16533

Description: uniex 4534 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 3902	. . 3 |- ( x = A -> U. x = U. A )
3	2	eleq1d 2300	. 2 |- ( x = A -> ( U. x e. _V <-> U. A e. _V ) )
4		bj-uniex2 16532	. . 3 |- E. y y = U. x
5	4	issetri 2812	. 2 |- U. x e. _V
6	1, 3, 5	vtocl 2858	1 |- U. A e. _V

Syntax hints:   = wceq 1397   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-un 4530  ax-bd0 16429  ax-bdex 16435  ax-bdel 16437  ax-bdsb 16438  ax-bdsep 16500

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  df-bdc 16457

This theorem is referenced by:  bj-uniexg  16534  bj-unex  16535