Theorem bj-uniex 15890

Description: uniex 4485 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 3859	. . 3 |- ( x = A -> U. x = U. A )
3	2	eleq1d 2274	. 2 |- ( x = A -> ( U. x e. _V <-> U. A e. _V ) )
4		bj-uniex2 15889	. . 3 |- E. y y = U. x
5	4	issetri 2781	. 2 |- U. x e. _V
6	1, 3, 5	vtocl 2827	1 |- U. A e. _V

Syntax hints:   = wceq 1373   e. wcel 2176   _Vcvv 2772   U.cuni 3850

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-un 4481  ax-bd0 15786  ax-bdex 15792  ax-bdel 15794  ax-bdsb 15795  ax-bdsep 15857

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-uni 3851  df-bdc 15814

This theorem is referenced by:  bj-uniexg  15891  bj-unex  15892