New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  minel GIF version

Theorem minel 3606
 Description: A minimum element of a class has no elements in common with the class. (Contributed by NM, 22-Jun-1994.)
Assertion
Ref Expression
minel ((A B (CB) = ) → ¬ A C)

Proof of Theorem minel
StepHypRef Expression
1 inelcm 3605 . . . . 5 ((A C A B) → (CB) ≠ )
21necon2bi 2562 . . . 4 ((CB) = → ¬ (A C A B))
3 imnan 411 . . . 4 ((A C → ¬ A B) ↔ ¬ (A C A B))
42, 3sylibr 203 . . 3 ((CB) = → (A C → ¬ A B))
54con2d 107 . 2 ((CB) = → (A B → ¬ A C))
65impcom 419 1 ((A B (CB) = ) → ¬ A C)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∩ cin 3208  ∅c0 3550 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 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator