Theorem bj-unexg 13496

Description: unexg 4403 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-unexg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)

Proof of Theorem bj-unexg

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uneq1 3254	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦))
2		eleq1 2220	. . 3 ⊢ ((𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦) → ((𝑥 ∪ 𝑦) ∈ V ↔ (𝐴 ∪ 𝑦) ∈ V))
3	1, 2	syl 14	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 ∪ 𝑦) ∈ V ↔ (𝐴 ∪ 𝑦) ∈ V))
4		uneq2 3255	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵))
5		eleq1 2220	. . 3 ⊢ ((𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵) → ((𝐴 ∪ 𝑦) ∈ V ↔ (𝐴 ∪ 𝐵) ∈ V))
6	4, 5	syl 14	. 2 ⊢ (𝑦 = 𝐵 → ((𝐴 ∪ 𝑦) ∈ V ↔ (𝐴 ∪ 𝐵) ∈ V))
7		vex 2715	. . 3 ⊢ 𝑥 ∈ V
8		vex 2715	. . 3 ⊢ 𝑦 ∈ V
9	7, 8	bj-unex 13494	. 2 ⊢ (𝑥 ∪ 𝑦) ∈ V
10	3, 6, 9	vtocl2g 2776	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1335   ∈ wcel 2128  Vcvv 2712   ∪ cun 3100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-pr 4169  ax-un 4393  ax-bd0 13388  ax-bdor 13391  ax-bdex 13394  ax-bdeq 13395  ax-bdel 13396  ax-bdsb 13397  ax-bdsep 13459

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567  df-uni 3773  df-bdc 13416

This theorem is referenced by:  bj-sucexg  13497