Theorem minel 3607

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 e. B /\ (C i^i B) = (/)) -> -. A e. C)

Proof of Theorem minel

Step	Hyp	Ref	Expression
1		inelcm 3606	. . . . 5 |- ((A e. C /\ A e. B) -> (C i^i B) =/= (/))
2	1	necon2bi 2563	. . . 4 |- ((C i^i B) = (/) -> -. (A e. C /\ A e. B))
3		imnan 411	. . . 4 |- ((A e. C -> -. A e. B) <-> -. (A e. C /\ A e. B))
4	2, 3	sylibr 203	. . 3 |- ((C i^i B) = (/) -> (A e. C -> -. A e. B))
5	4	con2d 107	. 2 |- ((C i^i B) = (/) -> (A e. B -> -. A e. C))
6	5	impcom 419	1 |- ((A e. B /\ (C i^i B) = (/)) -> -. A e. C)

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710   i^i 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: (None)