Theorem elex22 2786

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 2276	. . . 4 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
2		eleq1a 2276	. . . 4 ⊢ (𝐴 ∈ 𝐶 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐶))
3	1, 2	anim12ii 343	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)))
4	3	alrimiv 1896	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → ∀𝑥(𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)))
5		elisset 2785	. . 3 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴)
6	5	adantr 276	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → ∃𝑥 𝑥 = 𝐴)
7		exim 1621	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)))
8	4, 6, 7	sylc 62	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1370   = wceq 1372  ∃wex 1514   ∈ wcel 2175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-v 2773

This theorem is referenced by: (None)