Theorem inelcm 3606

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) → (B ∩ C) ≠ ∅)

Proof of Theorem inelcm

Step	Hyp	Ref	Expression
1		elin 3220	. 2 ⊢ (A ∈ (B ∩ C) ↔ (A ∈ B ∧ A ∈ C))
2		ne0i 3557	. 2 ⊢ (A ∈ (B ∩ C) → (B ∩ C) ≠ ∅)
3	1, 2	sylbir 204	1 ⊢ ((A ∈ B ∧ A ∈ C) → (B ∩ C) ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710   ≠ wne 2517   ∩ cin 3209  ∅c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is referenced by:  minel  3607