Theorem bj-unexg 10712

Description: unexg 4196 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 3119	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦))
2		eleq1 2141	. . 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 3120	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵))
5		eleq1 2141	. . 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 2604	. . 3 ⊢ 𝑥 ∈ V
8		vex 2604	. . 3 ⊢ 𝑦 ∈ V
9	7, 8	bj-unex 10710	. 2 ⊢ (𝑥 ∪ 𝑦) ∈ V
10	3, 6, 9	vtocl2g 2662	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284   ∈ wcel 1433  Vcvv 2601   ∪ cun 2971

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-pr 3964  ax-un 4188  ax-bd0 10604  ax-bdor 10607  ax-bdex 10610  ax-bdeq 10611  ax-bdel 10612  ax-bdsb 10613  ax-bdsep 10675

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-uni 3602  df-bdc 10632

This theorem is referenced by:  bj-sucexg  10713