Theorem unexb 2879

Description: Existence of union is equivalent to existence of its components.

Assertion

Ref	Expression
unexb	⊢ ((A ∈ V ⋀ B ∈ V) ↔ (A ∪ B) ∈ V)

Proof of Theorem unexb

Step	Hyp	Ref	Expression
1		uneq1 2180	. . . 4 ⊢ (x = A → (x ∪ y) = (A ∪ y))
2	1	eleq1d 1543	. . 3 ⊢ (x = A → ((x ∪ y) ∈ V ↔ (A ∪ y) ∈ V))
3		uneq2 2181	. . . 4 ⊢ (y = B → (A ∪ y) = (A ∪ B))
4	3	eleq1d 1543	. . 3 ⊢ (y = B → ((A ∪ y) ∈ V ↔ (A ∪ B) ∈ V))
5		visset 1816	. . . 4 ⊢ x ∈ V
6		visset 1816	. . . 4 ⊢ y ∈ V
7	5, 6	unex 2878	. . 3 ⊢ (x ∪ y) ∈ V
8	2, 4, 7	vtocl2g 1853	. 2 ⊢ ((A ∈ V ⋀ B ∈ V) → (A ∪ B) ∈ V)
9		ssun1 2196	. . . 4 ⊢ A ⊆ (A ∪ B)
10		ssexg 2726	. . . 4 ⊢ ((A ⊆ (A ∪ B) ⋀ (A ∪ B) ∈ V) → A ∈ V)
11	9, 10	mpan 697	. . 3 ⊢ ((A ∪ B) ∈ V → A ∈ V)
12		ssun2 2197	. . . 4 ⊢ B ⊆ (A ∪ B)
13		ssexg 2726	. . . 4 ⊢ ((B ⊆ (A ∪ B) ⋀ (A ∪ B) ∈ V) → B ∈ V)
14	12, 13	mpan 697	. . 3 ⊢ ((A ∪ B) ∈ V → B ∈ V)
15	11, 14	jca 288	. 2 ⊢ ((A ∪ B) ∈ V → (A ∈ V ⋀ B ∈ V))
16	8, 15	impbi 157	1 ⊢ ((A ∈ V ⋀ B ∈ V) ↔ (A ∪ B) ∈ V)

Syntax hints:   ↔ wb 146   ⋀ wa 223   = wceq 958   ∈ wcel 960  Vcvv 1814   ∪ cun 2048   ⊆ wss 2050

This theorem is referenced by:  unexg 2880  difex2 2883  sucexb 3054  unen 4440  fodomr 4489  cdavalt 4931  cnfilca 10562

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-uni 2508

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