Theorem elunii 1924

Description: Membership in class union.

Assertion

Ref	Expression
elunii	⊢ ((A ∈ B ∧ B ∈ C) → A ∈ ∪C)

Proof of Theorem elunii

Step	Hyp	Ref	Expression
1		eleq2 1150	. . . . 5 ⊢ (x = B → (A ∈ x ↔ A ∈ B))
2		eleq1 1149	. . . . 5 ⊢ (x = B → (x ∈ C ↔ B ∈ C))
3	1, 2	anbi12d 476	. . . 4 ⊢ (x = B → ((A ∈ x ∧ x ∈ C) ↔ (A ∈ B ∧ B ∈ C)))
4	3	cla4egv 1397	. . 3 ⊢ (B ∈ C → ((A ∈ B ∧ B ∈ C) → ∃x(A ∈ x ∧ x ∈ C)))
5	4	anabsi7 379	. 2 ⊢ ((A ∈ B ∧ B ∈ C) → ∃x(A ∈ x ∧ x ∈ C))
6		eluni 1922	. 2 ⊢ (A ∈ ∪C ↔ ∃x(A ∈ x ∧ x ∈ C))
7	5, 6	sylibr 175	1 ⊢ ((A ∈ B ∧ B ∈ C) → A ∈ ∪C)

Syntax hints:   → wi 2   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  ∪cuni 1919

This theorem is referenced by:  opeluu 1953  unon 2338  trcl 3489  aceq3 3556  suplem1pr 3955

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-v 1349  df-uni 1920

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