Theorem minel 3586

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 3585	. . . . 5 |- ( ( A e. C /\ A e. B ) -> ( C i^i B ) =/= (/) )
2	1	necon2bi 2475	. . . 4 |- ( ( C i^i B ) = (/) -> -. ( A e. C /\ A e. B ) )
3		imnan 701	. . . 4 |- ( ( A e. C -> -. A e. B ) <-> -. ( A e. C /\ A e. B ) )
4	2, 3	sylibr 134	. . 3 |- ( ( C i^i B ) = (/) -> ( A e. C -> -. A e. B ) )
5	4	con2d 633	. 2 |- ( ( C i^i B ) = (/) -> ( A e. B -> -. A e. C ) )
6	5	impcom 125	1 |- ( ( A e. B /\ ( C i^i B ) = (/) ) -> -. A e. C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   i^i cin 3219   (/)c0 3520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-v 2823  df-dif 3222  df-in 3226  df-nul 3521

This theorem is referenced by:  unfidisj  7223  hashunlem  11227  ccatval2  11349