Theorem minel 3419

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 3418	. . . . 5 |- ( ( A e. C /\ A e. B ) -> ( C i^i B ) =/= (/) )
2	1	necon2bi 2361	. . . 4 |- ( ( C i^i B ) = (/) -> -. ( A e. C /\ A e. B ) )
3		imnan 679	. . . 4 |- ( ( A e. C -> -. A e. B ) <-> -. ( A e. C /\ A e. B ) )
4	2, 3	sylibr 133	. . 3 |- ( ( C i^i B ) = (/) -> ( A e. C -> -. A e. B ) )
5	4	con2d 613	. 2 |- ( ( C i^i B ) = (/) -> ( A e. B -> -. A e. C ) )
6	5	impcom 124	1 |- ( ( A e. B /\ ( C i^i B ) = (/) ) -> -. A e. C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   i^i cin 3065   (/)c0 3358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-v 2683  df-dif 3068  df-in 3072  df-nul 3359

This theorem is referenced by:  unfidisj  6803  hashunlem  10543