Theorem bj-uniex 10866

Description: uniex 4200 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 3618	. . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
3	2	eleq1d 2148	. 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ V ↔ ∪ 𝐴 ∈ V))
4		bj-uniex2 10865	. . 3 ⊢ ∃𝑦 𝑦 = ∪ 𝑥
5	4	issetri 2609	. 2 ⊢ ∪ 𝑥 ∈ V
6	1, 3, 5	vtocl 2654	1 ⊢ ∪ 𝐴 ∈ V

Syntax hints:   = wceq 1285   ∈ wcel 1434  Vcvv 2602  ∪ cuni 3609

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-un 4196  ax-bd0 10762  ax-bdex 10768  ax-bdel 10770  ax-bdsb 10771  ax-bdsep 10833

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-uni 3610  df-bdc 10790

This theorem is referenced by:  bj-uniexg  10867  bj-unex  10868