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Theorem elex22 2563
 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
StepHypRef Expression
1 eleq1a 2106 . . . 4 (A B → (x = Ax B))
2 eleq1a 2106 . . . 4 (A 𝐶 → (x = Ax 𝐶))
31, 2anim12ii 325 . . 3 ((A B A 𝐶) → (x = A → (x B x 𝐶)))
43alrimiv 1751 . 2 ((A B A 𝐶) → x(x = A → (x B x 𝐶)))
5 elisset 2562 . . 3 (A Bx x = A)
65adantr 261 . 2 ((A B A 𝐶) → x x = A)
7 exim 1487 . 2 (x(x = A → (x B x 𝐶)) → (x x = Ax(x B x 𝐶)))
84, 6, 7sylc 56 1 ((A B A 𝐶) → x(x B x 𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1240   = wceq 1242  ∃wex 1378   ∈ wcel 1390 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553 This theorem is referenced by: (None)
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