Theorem minel 3305

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 3304	. . . . 5 |- ( ( A e. C /\ A e. B ) -> ( C i^i B ) =/= (/) )
2	1	necon2bi 2300	. . . 4 |- ( ( C i^i B ) = (/) -> -. ( A e. C /\ A e. B ) )
3		imnan 656	. . . 4 |- ( ( A e. C -> -. A e. B ) <-> -. ( A e. C /\ A e. B ) )
4	2, 3	sylibr 132	. . 3 |- ( ( C i^i B ) = (/) -> ( A e. C -> -. A e. B ) )
5	4	con2d 586	. 2 |- ( ( C i^i B ) = (/) -> ( A e. B -> -. A e. C ) )
6	5	impcom 123	1 |- ( ( A e. B /\ ( C i^i B ) = (/) ) -> -. A e. C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   i^i cin 2972   (/)c0 3251

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-v 2603  df-dif 2975  df-in 2979  df-nul 3252

This theorem is referenced by: (None)