Theorem bj-uniex 13104

Description: uniex 4354 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 3740	. . 3 |- ( x = A -> U. x = U. A )
3	2	eleq1d 2206	. 2 |- ( x = A -> ( U. x e. _V <-> U. A e. _V ) )
4		bj-uniex2 13103	. . 3 |- E. y y = U. x
5	4	issetri 2690	. 2 |- U. x e. _V
6	1, 3, 5	vtocl 2735	1 |- U. A e. _V

Syntax hints:   = wceq 1331   e. wcel 1480   _Vcvv 2681   U.cuni 3731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-un 4350  ax-bd0 13000  ax-bdex 13006  ax-bdel 13008  ax-bdsb 13009  ax-bdsep 13071

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683  df-uni 3732  df-bdc 13028

This theorem is referenced by:  bj-uniexg  13105  bj-unex  13106