Theorem uniexb 1962

Description: The Axiom of Union and its converse. A class is a set iff its union is a set.

Assertion

Ref	Expression
uniexb	⊢ (A ∈ V ↔ ∪A ∈ V)

Proof of Theorem uniexb

Step	Hyp	Ref	Expression
1		uniexg 1948	. 2 ⊢ (A ∈ V → ∪A ∈ V)
2		pwuni 1961	. . 3 ⊢ A ⊆ ℘∪A
3		pwexg 1807	. . . 4 ⊢ (∪A ∈ V → ℘∪A ∈ V)
4		ssexg 1702	. . . 4 ⊢ (℘∪A ∈ V → (A ⊆ ℘∪A → A ∈ V))
5	3, 4	syl 12	. . 3 ⊢ (∪A ∈ V → (A ⊆ ℘∪A → A ∈ V))
6	2, 5	mpi 44	. 2 ⊢ (∪A ∈ V → A ∈ V)
7	1, 6	impbi 139	1 ⊢ (A ∈ V ↔ ∪A ∈ V)

Syntax hints:   → wi 2   ↔ wb 127   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ℘cpw 1798  ∪cuni 1919

This theorem is referenced by:  pwexb 1963

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-in 1491  df-ss 1492  df-pw 1799  df-uni 1920

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