Theorem unexb 1950

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 1605	. . . 4 ⊢ (x = A → (x ∪ y) = (A ∪ y))
2	1	eleq1d 1155	. . 3 ⊢ (x = A → ((x ∪ y) ∈ V ↔ (A ∪ y) ∈ V))
3		uneq2 1606	. . . 4 ⊢ (y = B → (A ∪ y) = (A ∪ B))
4	3	eleq1d 1155	. . 3 ⊢ (y = B → ((A ∪ y) ∈ V ↔ (A ∪ B) ∈ V))
5		visset 1350	. . . 4 ⊢ x ∈ V
6		visset 1350	. . . 4 ⊢ y ∈ V
7	5, 6	unex 1949	. . 3 ⊢ (x ∪ y) ∈ V
8	2, 4, 7	vtocl2g 1386	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → (A ∪ B) ∈ V)
9		ssun1 1621	. . . 4 ⊢ A ⊆ (A ∪ B)
10		ssexg 1702	. . . 4 ⊢ ((A ∪ B) ∈ V → (A ⊆ (A ∪ B) → A ∈ V))
11	9, 10	mpi 44	. . 3 ⊢ ((A ∪ B) ∈ V → A ∈ V)
12		ssun2 1622	. . . 4 ⊢ B ⊆ (A ∪ B)
13		ssexg 1702	. . . 4 ⊢ ((A ∪ B) ∈ V → (B ⊆ (A ∪ B) → B ∈ V))
14	12, 13	mpi 44	. . 3 ⊢ ((A ∪ B) ∈ V → B ∈ V)
15	11, 14	jca 236	. 2 ⊢ ((A ∪ B) ∈ V → (A ∈ V ∧ B ∈ V))
16	8, 15	impbi 139	1 ⊢ ((A ∈ V ∧ B ∈ V) ↔ (A ∪ B) ∈ V)

Syntax hints:   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∪ cun 1485   ⊆ wss 1487

This theorem is referenced by:  difex2 1951  sucexb 2301  unen 3338  alephprc 3698  cdavalt 3716

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-uni 1920

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