Theorem elex22 3506

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 2836	. . . 4 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
2		eleq1a 2836	. . . 4 ⊢ (𝐴 ∈ 𝐶 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐶))
3	1, 2	anim12ii 618	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)))
4	3	alrimiv 1927	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → ∀𝑥(𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)))
5		elissetv 2822	. . 3 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴)
6	5	adantr 480	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → ∃𝑥 𝑥 = 𝐴)
7		exim 1834	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)))
8	4, 6, 7	sylc 65	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2729  df-clel 2816

This theorem is referenced by:  en3lplem1VD  44863