Theorem bj-uniex 15147

Description: uniex 4455 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 3833	. . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
3	2	eleq1d 2258	. 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ V ↔ ∪ 𝐴 ∈ V))
4		bj-uniex2 15146	. . 3 ⊢ ∃𝑦 𝑦 = ∪ 𝑥
5	4	issetri 2761	. 2 ⊢ ∪ 𝑥 ∈ V
6	1, 3, 5	vtocl 2806	1 ⊢ ∪ 𝐴 ∈ V

Syntax hints:   = wceq 1364   ∈ wcel 2160  Vcvv 2752  ∪ cuni 3824

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-un 4451  ax-bd0 15043  ax-bdex 15049  ax-bdel 15051  ax-bdsb 15052  ax-bdsep 15114

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-uni 3825  df-bdc 15071

This theorem is referenced by:  bj-uniexg  15148  bj-unex  15149