Theorem unexbOLD 7696

Description: Obsolete version of unexb 7695 as of 21-Jul-2025. (Contributed by NM, 11-Jun-1998.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
unexbOLD	⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V)

Proof of Theorem unexbOLD

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

Step	Hyp	Ref	Expression
1		uneq1 4114	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦))
2	1	eleq1d 2822	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∪ 𝑦) ∈ V ↔ (𝐴 ∪ 𝑦) ∈ V))
3		uneq2 4115	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵))
4	3	eleq1d 2822	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ∪ 𝑦) ∈ V ↔ (𝐴 ∪ 𝐵) ∈ V))
5		vex 3445	. . . 4 ⊢ 𝑥 ∈ V
6		vex 3445	. . . 4 ⊢ 𝑦 ∈ V
7	5, 6	unex 7692	. . 3 ⊢ (𝑥 ∪ 𝑦) ∈ V
8	2, 4, 7	vtocl2g 3530	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
9		ssun1 4131	. . . 4 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
10		ssexg 5269	. . . 4 ⊢ ((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐴 ∈ V)
11	9, 10	mpan 691	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → 𝐴 ∈ V)
12		ssun2 4132	. . . 4 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
13		ssexg 5269	. . . 4 ⊢ ((𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐵 ∈ V)
14	12, 13	mpan 691	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → 𝐵 ∈ V)
15	11, 14	jca 511	. 2 ⊢ ((𝐴 ∪ 𝐵) ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
16	8, 15	impbii 209	1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3441   ∪ cun 3900   ⊆ wss 3902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-sn 4582  df-pr 4584  df-uni 4865

This theorem is referenced by: (None)