Theorem unexb 7766

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 7762	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
2		ssun1 4188	. . . 4 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
3		ssexg 5329	. . . 4 ⊢ ((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐴 ∈ V)
4	2, 3	mpan 690	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → 𝐴 ∈ V)
5		ssun2 4189	. . . 4 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
6		ssexg 5329	. . . 4 ⊢ ((𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐵 ∈ V)
7	5, 6	mpan 690	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → 𝐵 ∈ V)
8	4, 7	jca 511	. 2 ⊢ ((𝐴 ∪ 𝐵) ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
9	1, 8	impbii 209	1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V)

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2106  Vcvv 3478   ∪ cun 3961   ⊆ wss 3963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-sn 4632  df-pr 4634  df-uni 4913

This theorem is referenced by:  unexgOLD  7768  sucexb  7824  fodomr  9167  fsuppun  9425  fsuppunbi  9427  djuexb  9947  bj-tagex  36970