Theorem bj-uniex 13286

Description: uniex 4367 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 3753	. . 3 |- ( x = A -> U. x = U. A )
3	2	eleq1d 2209	. 2 |- ( x = A -> ( U. x e. _V <-> U. A e. _V ) )
4		bj-uniex2 13285	. . 3 |- E. y y = U. x
5	4	issetri 2698	. 2 |- U. x e. _V
6	1, 3, 5	vtocl 2743	1 |- U. A e. _V

Syntax hints:   = wceq 1332   e. wcel 1481   _Vcvv 2689   U.cuni 3744

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-un 4363  ax-bd0 13182  ax-bdex 13188  ax-bdel 13190  ax-bdsb 13191  ax-bdsep 13253

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-uni 3745  df-bdc 13210

This theorem is referenced by:  bj-uniexg  13287  bj-unex  13288