Theorem unexb 4371

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 3228	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦))
2	1	eleq1d 2209	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∪ 𝑦) ∈ V ↔ (𝐴 ∪ 𝑦) ∈ V))
3		uneq2 3229	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵))
4	3	eleq1d 2209	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ∪ 𝑦) ∈ V ↔ (𝐴 ∪ 𝐵) ∈ V))
5		vex 2692	. . . 4 ⊢ 𝑥 ∈ V
6		vex 2692	. . . 4 ⊢ 𝑦 ∈ V
7	5, 6	unex 4370	. . 3 ⊢ (𝑥 ∪ 𝑦) ∈ V
8	2, 4, 7	vtocl2g 2753	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
9		ssun1 3244	. . . 4 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
10		ssexg 4075	. . . 4 ⊢ ((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐴 ∈ V)
11	9, 10	mpan 421	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → 𝐴 ∈ V)
12		ssun2 3245	. . . 4 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
13		ssexg 4075	. . . 4 ⊢ ((𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐵 ∈ V)
14	12, 13	mpan 421	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → 𝐵 ∈ V)
15	11, 14	jca 304	. 2 ⊢ ((𝐴 ∪ 𝐵) ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
16	8, 15	impbii 125	1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V)

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 1481  Vcvv 2689   ∪ cun 3074   ⊆ wss 3076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-sn 3538  df-pr 3539  df-uni 3745

This theorem is referenced by:  unexg  4372  sucexb  4421  frecabex  6303  djuexb  6937