Theorem elex22 2870

Description: If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.)

Assertion

Ref	Expression
elex22	⊢ ((A ∈ B ∧ A ∈ C) → ∃x(x ∈ B ∧ x ∈ C))

Distinct variable groups:   x,A   x,B   x,C

Proof of Theorem elex22

Step	Hyp	Ref	Expression
1		eleq1a 2422	. . . 4 ⊢ (A ∈ B → (x = A → x ∈ B))
2		eleq1a 2422	. . . 4 ⊢ (A ∈ C → (x = A → x ∈ C))
3	1, 2	anim12ii 553	. . 3 ⊢ ((A ∈ B ∧ A ∈ C) → (x = A → (x ∈ B ∧ x ∈ C)))
4	3	alrimiv 1631	. 2 ⊢ ((A ∈ B ∧ A ∈ C) → ∀x(x = A → (x ∈ B ∧ x ∈ C)))
5		elisset 2869	. . 3 ⊢ (A ∈ B → ∃x x = A)
6	5	adantr 451	. 2 ⊢ ((A ∈ B ∧ A ∈ C) → ∃x x = A)
7		exim 1575	. 2 ⊢ (∀x(x = A → (x ∈ B ∧ x ∈ C)) → (∃x x = A → ∃x(x ∈ B ∧ x ∈ C)))
8	4, 6, 7	sylc 56	1 ⊢ ((A ∈ B ∧ A ∈ C) → ∃x(x ∈ B ∧ x ∈ C))

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710

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-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861

This theorem is referenced by: (None)