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Theorem unexb 7687
Description: Existence of union is equivalent to existence of its components. (Contributed by NM, 11-Jun-1998.)
Assertion
Ref Expression
unexb ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴𝐵) ∈ V)

Proof of Theorem unexb
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uneq1 4121 . . . 4 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
21eleq1d 2823 . . 3 (𝑥 = 𝐴 → ((𝑥𝑦) ∈ V ↔ (𝐴𝑦) ∈ V))
3 uneq2 4122 . . . 4 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
43eleq1d 2823 . . 3 (𝑦 = 𝐵 → ((𝐴𝑦) ∈ V ↔ (𝐴𝐵) ∈ V))
5 vex 3452 . . . 4 𝑥 ∈ V
6 vex 3452 . . . 4 𝑦 ∈ V
75, 6unex 7685 . . 3 (𝑥𝑦) ∈ V
82, 4, 7vtocl2g 3534 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
9 ssun1 4137 . . . 4 𝐴 ⊆ (𝐴𝐵)
10 ssexg 5285 . . . 4 ((𝐴 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ V) → 𝐴 ∈ V)
119, 10mpan 689 . . 3 ((𝐴𝐵) ∈ V → 𝐴 ∈ V)
12 ssun2 4138 . . . 4 𝐵 ⊆ (𝐴𝐵)
13 ssexg 5285 . . . 4 ((𝐵 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ V) → 𝐵 ∈ V)
1412, 13mpan 689 . . 3 ((𝐴𝐵) ∈ V → 𝐵 ∈ V)
1511, 14jca 513 . 2 ((𝐴𝐵) ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
168, 15impbii 208 1 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wcel 2107  Vcvv 3448  cun 3913  wss 3915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-sn 4592  df-pr 4594  df-uni 4871
This theorem is referenced by:  unexg  7688  sucexb  7744  fodomr  9079  fsuppun  9331  fsuppunbi  9333  djuexb  9852  bj-tagex  35487
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