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Theorem unexb 7750
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 4155 . . . 4 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
21eleq1d 2814 . . 3 (𝑥 = 𝐴 → ((𝑥𝑦) ∈ V ↔ (𝐴𝑦) ∈ V))
3 uneq2 4156 . . . 4 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
43eleq1d 2814 . . 3 (𝑦 = 𝐵 → ((𝐴𝑦) ∈ V ↔ (𝐴𝐵) ∈ V))
5 vex 3475 . . . 4 𝑥 ∈ V
6 vex 3475 . . . 4 𝑦 ∈ V
75, 6unex 7748 . . 3 (𝑥𝑦) ∈ V
82, 4, 7vtocl2g 3560 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
9 ssun1 4172 . . . 4 𝐴 ⊆ (𝐴𝐵)
10 ssexg 5323 . . . 4 ((𝐴 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ V) → 𝐴 ∈ V)
119, 10mpan 689 . . 3 ((𝐴𝐵) ∈ V → 𝐴 ∈ V)
12 ssun2 4173 . . . 4 𝐵 ⊆ (𝐴𝐵)
13 ssexg 5323 . . . 4 ((𝐵 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ V) → 𝐵 ∈ V)
1412, 13mpan 689 . . 3 ((𝐴𝐵) ∈ V → 𝐵 ∈ V)
1511, 14jca 511 . 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 395   = wceq 1534  wcel 2099  Vcvv 3471  cun 3945  wss 3947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-sn 4630  df-pr 4632  df-uni 4909
This theorem is referenced by:  unexg  7751  sucexb  7807  fodomr  9152  fsuppun  9410  fsuppunbi  9412  djuexb  9932  bj-tagex  36466
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