Theorem eluni2 1923

Description: Membership in class union. Restricted quantifier version.

Assertion

Ref	Expression
eluni2	⊢ (A ∈ ∪B ↔ ∃x ∈ B A ∈ x)

Distinct variable group(s):   x,A   x,B

Proof of Theorem eluni2

Step	Hyp	Ref	Expression
1		exancom 736	. 2 ⊢ (∃x(A ∈ x ∧ x ∈ B) ↔ ∃x(x ∈ B ∧ A ∈ x))
2		eluni 1922	. 2 ⊢ (A ∈ ∪B ↔ ∃x(A ∈ x ∧ x ∈ B))
3		df-rex 1206	. 2 ⊢ (∃x ∈ B A ∈ x ↔ ∃x(x ∈ B ∧ A ∈ x))
4	1, 2, 3	3bitr4 158	1 ⊢ (A ∈ ∪B ↔ ∃x ∈ B A ∈ x)

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678   ∈ wcel 1092  ∃wrex 1202  ∪cuni 1919

This theorem is referenced by:  uni0b 1939  iununi 2037  ssorduni 2249  unon 2338  reluni 2493  cnvuni 2521  chfnrn 2885  rankuni 3533  cflim 3704

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-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-uni 1920

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