Theorem bj-unexg 15651

Description: unexg 4479 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 3311	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦))
2		eleq1 2259	. . 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 3312	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵))
5		eleq1 2259	. . 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 2766	. . 3 ⊢ 𝑥 ∈ V
8		vex 2766	. . 3 ⊢ 𝑦 ∈ V
9	7, 8	bj-unex 15649	. 2 ⊢ (𝑥 ∪ 𝑦) ∈ V
10	3, 6, 9	vtocl2g 2828	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167  Vcvv 2763   ∪ cun 3155

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-pr 4243  ax-un 4469  ax-bd0 15543  ax-bdor 15546  ax-bdex 15549  ax-bdeq 15550  ax-bdel 15551  ax-bdsb 15552  ax-bdsep 15614

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-un 3161  df-sn 3629  df-pr 3630  df-uni 3841  df-bdc 15571

This theorem is referenced by:  bj-sucexg  15652