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Theorem unexb 7290
 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 4023 . . . 4 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
21eleq1d 2850 . . 3 (𝑥 = 𝐴 → ((𝑥𝑦) ∈ V ↔ (𝐴𝑦) ∈ V))
3 uneq2 4024 . . . 4 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
43eleq1d 2850 . . 3 (𝑦 = 𝐵 → ((𝐴𝑦) ∈ V ↔ (𝐴𝐵) ∈ V))
5 vex 3418 . . . 4 𝑥 ∈ V
6 vex 3418 . . . 4 𝑦 ∈ V
75, 6unex 7288 . . 3 (𝑥𝑦) ∈ V
82, 4, 7vtocl2g 3490 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
9 ssun1 4039 . . . 4 𝐴 ⊆ (𝐴𝐵)
10 ssexg 5084 . . . 4 ((𝐴 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ V) → 𝐴 ∈ V)
119, 10mpan 677 . . 3 ((𝐴𝐵) ∈ V → 𝐴 ∈ V)
12 ssun2 4040 . . . 4 𝐵 ⊆ (𝐴𝐵)
13 ssexg 5084 . . . 4 ((𝐵 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ V) → 𝐵 ∈ V)
1412, 13mpan 677 . . 3 ((𝐴𝐵) ∈ V → 𝐵 ∈ V)
1511, 14jca 504 . 2 ((𝐴𝐵) ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
168, 15impbii 201 1 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴𝐵) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2050  Vcvv 3415   ∪ cun 3829   ⊆ wss 3831 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pr 5187  ax-un 7281 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-rex 3094  df-v 3417  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-sn 4443  df-pr 4445  df-uni 4714 This theorem is referenced by:  unexg  7291  sucexb  7342  fodomr  8466  fsuppun  8649  fsuppunbi  8651  djuexb  9134  cdaval  9391  bj-tagex  33817
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