Theorem bj-uniex 15573

Description: uniex 4473 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 3849	. . 3 |- ( x = A -> U. x = U. A )
3	2	eleq1d 2265	. 2 |- ( x = A -> ( U. x e. _V <-> U. A e. _V ) )
4		bj-uniex2 15572	. . 3 |- E. y y = U. x
5	4	issetri 2772	. 2 |- U. x e. _V
6	1, 3, 5	vtocl 2818	1 |- U. A e. _V

Syntax hints:   = wceq 1364   e. wcel 2167   _Vcvv 2763   U.cuni 3840

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-un 4469  ax-bd0 15469  ax-bdex 15475  ax-bdel 15477  ax-bdsb 15478  ax-bdsep 15540

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-uni 3841  df-bdc 15497

This theorem is referenced by:  bj-uniexg  15574  bj-unex  15575