Theorem minel 3477

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 3476	. . . . 5 |- ( ( A e. C /\ A e. B ) -> ( C i^i B ) =/= (/) )
2	1	necon2bi 2396	. . . 4 |- ( ( C i^i B ) = (/) -> -. ( A e. C /\ A e. B ) )
3		imnan 686	. . . 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 620	. 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 1349   e. wcel 2142   i^i cin 3121   (/)c0 3415

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 610  ax-in2 611  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-ext 2153

This theorem depends on definitions:  df-bi 116  df-tru 1352  df-nf 1455  df-sb 1757  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ne 2342  df-v 2733  df-dif 3124  df-in 3128  df-nul 3416

This theorem is referenced by:  unfidisj  6903  hashunlem  10743