Theorem bj-uniex 16572

Description: uniex 4536 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 3903	. . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
3	2	eleq1d 2299	. 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ V ↔ ∪ 𝐴 ∈ V))
4		bj-uniex2 16571	. . 3 ⊢ ∃𝑦 𝑦 = ∪ 𝑥
5	4	issetri 2811	. 2 ⊢ ∪ 𝑥 ∈ V
6	1, 3, 5	vtocl 2857	1 ⊢ ∪ 𝐴 ∈ V

Syntax hints:   = wceq 1397   ∈ wcel 2201  Vcvv 2801  ∪ cuni 3894

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-un 4532  ax-bd0 16468  ax-bdex 16474  ax-bdel 16476  ax-bdsb 16477  ax-bdsep 16539

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-rex 2515  df-v 2803  df-uni 3895  df-bdc 16496

This theorem is referenced by:  bj-uniexg  16573  bj-unex  16574