Theorem inelcm 1742

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

Assertion

Ref	Expression
inelcm	⊢ ((A ∈ B ∧ A ∈ C) → ¬ (B ∩ C) = ∅)

Proof of Theorem inelcm

Step	Hyp	Ref	Expression
1		elin 1635	. 2 ⊢ (A ∈ (B ∩ C) ↔ (A ∈ B ∧ A ∈ C))
2		n0i 1712	. 2î⊢ (A ∈ (B ∩ C) → ¬ (B ∩ C) = ∅)
3	1, 2	sylbir 176	1 ⊢ ((A ∈ B ∧ A ∈ C) → ¬ (B ∩ C) = ∅)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092   ∩ cin 1486  ∅c0 1707

This theorem is referenced by:  minel 1743  fr2nr 2177  fr3nr 2178  cplem1 3545

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-in 1491  df-nul 1708

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