Theorem minel 2377

Description: A minimum element of a class has no elements in common with the class.

Assertion

Ref	Expression
minel	|- ((A e. B /\ (C i^i B) = (/)) -> -. A e. C)

Proof of Theorem minel

Step	Hyp	Ref	Expression
1		inelcm 2376	. . . . 5 |- ((A e. C /\ A e. B) -> (C i^i B) =/= (/))
2	1	necon2bi 1655	. . . 4 |- ((C i^i B) = (/) -> -. (A e. C /\ A e. B))
3		imnan 240	. . . 4 |- ((A e. C -> -. A e. B) <-> -. (A e. C /\ A e. B))
4	2, 3	sylibr 198	. . 3 |- ((C i^i B) = (/) -> (A e. C -> -. A e. B))
5	4	con2d 91	. 2 |- ((C i^i B) = (/) -> (A e. B -> -. A e. C))
6	5	impcom 349	1 |- ((A e. B /\ (C i^i B) = (/)) -> -. A e. C)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 221   = wceq 992   e. wcel 994   i^i cin 2098  (/)c0 2332

This theorem is referenced by:  peano5 3241  aceq5 4886

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-v 1858  df-dif 2101  df-in 2103  df-nul 2333

Copyright terms: Public domain