Theorem uniexg 1948

Description: The ZF Axiom of Union in class notation, in the form of a theorem instead of an inference. We use the antecedent A ∈ B instead of A ∈ V to make the theorem more general and thus shorten some proofs; obviously V is one possibility for B.

Assertion

Ref	Expression
uniexg	⊢ (A ∈ B → ∪A ∈ V)

Proof of Theorem uniexg

Step	Hyp	Ref	Expression
1		unieq 1927	. . 3 ⊢ (x = A → ∪x = ∪A)
2	1	eleq1d 1155	. 2 ⊢ (x = A → (∪x ∈ V ↔ ∪A ∈ V))
3		visset 1350	. . 3 ⊢ x ∈ V
4	3	uniex 1947	. 2 ⊢ ∪x ∈ V
5	2, 4	vtoclg 1383	1 ⊢ (A ∈ B → ∪A ∈ V)

Syntax hints:   → wi 2   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ∪cuni 1919

This theorem is referenced by:  euuni 1954  uniexb 1962  onunit 2250  dmexg 2551  rnexg 2569  tz7.44lem1 2965  carduni 3664  cardprc 3667  suplem2pr 3956  pjvalt 5246

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

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-uni 1920

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