Theorem bj-uniex 13952

Description: uniex 4422 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 3805	. . 3 |- ( x = A -> U. x = U. A )
3	2	eleq1d 2239	. 2 |- ( x = A -> ( U. x e. _V <-> U. A e. _V ) )
4		bj-uniex2 13951	. . 3 |- E. y y = U. x
5	4	issetri 2739	. 2 |- U. x e. _V
6	1, 3, 5	vtocl 2784	1 |- U. A e. _V

Syntax hints:   = wceq 1348   e. wcel 2141   _Vcvv 2730   U.cuni 3796

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-un 4418  ax-bd0 13848  ax-bdex 13854  ax-bdel 13856  ax-bdsb 13857  ax-bdsep 13919

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-uni 3797  df-bdc 13876

This theorem is referenced by:  bj-uniexg  13953  bj-unex  13954