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Theorem inelcm 3259
 Description: The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994.)
Assertion
Ref Expression
inelcm ((A B A 𝐶) → (B𝐶) ≠ ∅)

Proof of Theorem inelcm
StepHypRef Expression
1 elin 3103 . 2 (A (B𝐶) ↔ (A B A 𝐶))
2 ne0i 3207 . 2 (A (B𝐶) → (B𝐶) ≠ ∅)
31, 2sylbir 125 1 ((A B A 𝐶) → (B𝐶) ≠ ∅)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1374   ≠ wne 2186   ∩ cin 2893  ∅c0 3201 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-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-v 2537  df-dif 2897  df-in 2901  df-nul 3202 This theorem is referenced by:  minel  3260
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