Theorem elunii 3897

Description: Membership in class union. (Contributed by NM, 24-Mar-1995.)

Assertion

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

Proof of Theorem elunii

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2414	. . . . 5 ⊢ (x = B → (A ∈ x ↔ A ∈ B))
2		eleq1 2413	. . . . 5 ⊢ (x = B → (x ∈ C ↔ B ∈ C))
3	1, 2	anbi12d 691	. . . 4 ⊢ (x = B → ((A ∈ x ∧ x ∈ C) ↔ (A ∈ B ∧ B ∈ C)))
4	3	spcegv 2941	. . 3 ⊢ (B ∈ C → ((A ∈ B ∧ B ∈ C) → ∃x(A ∈ x ∧ x ∈ C)))
5	4	anabsi7 792	. 2 ⊢ ((A ∈ B ∧ B ∈ C) → ∃x(A ∈ x ∧ x ∈ C))
6		eluni 3895	. 2 ⊢ (A ∈ ∪C ↔ ∃x(A ∈ x ∧ x ∈ C))
7	5, 6	sylibr 203	1 ⊢ ((A ∈ B ∧ B ∈ C) → A ∈ ∪C)

Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∪cuni 3892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-uni 3893

This theorem is referenced by:  ssuni  3914  unipw  4118  nnadjoin  4521  sfinltfin  4536  nulnnn  4557