Theorem unexb 7725

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) ↔ (𝐴 ∪ 𝐵) ∈ V)

Proof of Theorem unexb

Step	Hyp	Ref	Expression
1		unexg 7721	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
2		ssun1 4128	. . . 4 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
3		ssexg 5276	. . . 4 ⊢ ((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐴 ∈ V)
4	2, 3	mpan 700	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → 𝐴 ∈ V)
5		ssun2 4129	. . . 4 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
6		ssexg 5276	. . . 4 ⊢ ((𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐵 ∈ V)
7	5, 6	mpan 700	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → 𝐵 ∈ V)
8	4, 7	jca 519	. 2 ⊢ ((𝐴 ∪ 𝐵) ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
9	1, 8	impbii 211	1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V)

Syntax hints:   ↔ wb 208   ∧ wa 399   ∈ wcel 2141  Vcvv 3453   ∪ cun 3900   ⊆ wss 3902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-un 3907  df-in 3909  df-ss 3919  df-sn 4580  df-pr 4582  df-uni 4863

This theorem is referenced by:  unexgOLD  7727  sucexb  7782  fodomr  9094  fsuppun  9327  fsuppunbi  9329  djuexb  9861  bj-tagex  37433