Theorem bj-uniex 14209

Description: uniex 4431 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-uniex.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
bj-uniex	⊢ ∪ 𝐴 ∈ 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 ∈ V
2		unieq 3814	. . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
3	2	eleq1d 2244	. 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ V ↔ ∪ 𝐴 ∈ V))
4		bj-uniex2 14208	. . 3 ⊢ ∃𝑦 𝑦 = ∪ 𝑥
5	4	issetri 2744	. 2 ⊢ ∪ 𝑥 ∈ V
6	1, 3, 5	vtocl 2789	1 ⊢ ∪ 𝐴 ∈ V

Syntax hints:   = wceq 1353   ∈ wcel 2146  Vcvv 2735  ∪ cuni 3805

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-un 4427  ax-bd0 14105  ax-bdex 14111  ax-bdel 14113  ax-bdsb 14114  ax-bdsep 14176

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rex 2459  df-v 2737  df-uni 3806  df-bdc 14133

This theorem is referenced by:  bj-uniexg  14210  bj-unex  14211