Theorem minel 2320

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 2319	. . . . 5 |- ((A e. C /\ A e. B) -> (C i^i B) =/= (/))
2	1	necon2bi 1609	. . . 4 |- ((C i^i B) = (/) -> -. (A e. C /\ A e. B))
3		imnan 242	. . . 4 |- ((A e. C -> -. A e. B) <-> -. (A e. C /\ A e. B))
4	2, 3	sylibr 200	. . 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 351	1 |- ((A e. B /\ (C i^i B) = (/)) -> -. A e. C)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 954   e. wcel 956   i^i cin 2042  (/)c0 2276

This theorem is referenced by:  peano5 3148  aceq5 4720

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-in 2047  df-nul 2277

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