Theorem bj-uniex 16052

Description: uniex 4502 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 3873	. . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
3	2	eleq1d 2276	. 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ V ↔ ∪ 𝐴 ∈ V))
4		bj-uniex2 16051	. . 3 ⊢ ∃𝑦 𝑦 = ∪ 𝑥
5	4	issetri 2786	. 2 ⊢ ∪ 𝑥 ∈ V
6	1, 3, 5	vtocl 2832	1 ⊢ ∪ 𝐴 ∈ V

Syntax hints:   = wceq 1373   ∈ wcel 2178  Vcvv 2776  ∪ cuni 3864

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-un 4498  ax-bd0 15948  ax-bdex 15954  ax-bdel 15956  ax-bdsb 15957  ax-bdsep 16019

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-v 2778  df-uni 3865  df-bdc 15976

This theorem is referenced by:  bj-uniexg  16053  bj-unex  16054