Theorem inelcm 2327

Description: The intersection of classes with a common member is nonempty.

Assertion

Ref	Expression
inelcm	|- ((A e. B /\ A e. C) -> (B i^i C) =/= (/))

Proof of Theorem inelcm

Step	Hyp	Ref	Expression
1		elin 2210	. 2 |- (A e. (B i^i C) <-> (A e. B /\ A e. C))
2		ne0i 2289	. 2 |- (A e. (B i^i C) -> (B i^i C) =/= (/))
3	1, 2	sylbir 201	1 |- ((A e. B /\ A e. C) -> (B i^i C) =/= (/))

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 960   =/= wne 1588   i^i cin 2049  (/)c0 2283

This theorem is referenced by:  minel 2328  fr2nr 2931  fr3nr 2932  cplem1 4730  metelcls 7962  uninqs 10436

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-in 2054  df-nul 2284

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