Theorem minel 3558

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 3557	. . . . 5 |- ( ( A e. C /\ A e. B ) -> ( C i^i B ) =/= (/) )
2	1	necon2bi 2458	. . . 4 |- ( ( C i^i B ) = (/) -> -. ( A e. C /\ A e. B ) )
3		imnan 697	. . . 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 629	. 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 1398   e. wcel 2202   i^i cin 3200   (/)c0 3496

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-v 2805  df-dif 3203  df-in 3207  df-nul 3497

This theorem is referenced by:  unfidisj  7157  hashunlem  11111  ccatval2  11222