Theorem unexb 7742

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 7738	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
2		ssun1 4139	. . . 4 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
3		ssexg 5291	. . . 4 ⊢ ((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐴 ∈ V)
4	2, 3	mpan 702	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → 𝐴 ∈ V)
5		ssun2 4140	. . . 4 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
6		ssexg 5291	. . . 4 ⊢ ((𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐵 ∈ V)
7	5, 6	mpan 702	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → 𝐵 ∈ V)
8	4, 7	jca 520	. 2 ⊢ ((𝐴 ∪ 𝐵) ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
9	1, 8	impbii 212	1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V)

Syntax hints:   ↔ wb 209   ∧ wa 400   ∈ wcel 2149  Vcvv 3463   ∪ cun 3911   ⊆ wss 3913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-un 3918  df-in 3920  df-ss 3930  df-sn 4592  df-pr 4594  df-uni 4874

This theorem is referenced by:  unexgOLD  7744  sucexb  7799  fodomr  9112  fsuppun  9343  fsuppunbi  9345  djuexb  9891  bj-tagex  37507