Theorem unexb 4477

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

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uneq1 3310	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦))
2	1	eleq1d 2265	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∪ 𝑦) ∈ V ↔ (𝐴 ∪ 𝑦) ∈ V))
3		uneq2 3311	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵))
4	3	eleq1d 2265	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ∪ 𝑦) ∈ V ↔ (𝐴 ∪ 𝐵) ∈ V))
5		vex 2766	. . . 4 ⊢ 𝑥 ∈ V
6		vex 2766	. . . 4 ⊢ 𝑦 ∈ V
7	5, 6	unex 4476	. . 3 ⊢ (𝑥 ∪ 𝑦) ∈ V
8	2, 4, 7	vtocl2g 2828	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
9		ssun1 3326	. . . 4 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
10		ssexg 4172	. . . 4 ⊢ ((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐴 ∈ V)
11	9, 10	mpan 424	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → 𝐴 ∈ V)
12		ssun2 3327	. . . 4 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
13		ssexg 4172	. . . 4 ⊢ ((𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐵 ∈ V)
14	12, 13	mpan 424	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → 𝐵 ∈ V)
15	11, 14	jca 306	. 2 ⊢ ((𝐴 ∪ 𝐵) ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
16	8, 15	impbii 126	1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V)

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167  Vcvv 2763   ∪ cun 3155   ⊆ wss 3157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-pr 3629  df-uni 3840

This theorem is referenced by:  unexg  4478  sucexb  4533  frecabex  6456  djuexb  7110