Theorem minel 2929

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 2928	. . . . 5 |- ((A e. C /\ A e. B) -> (C i^i B) =/= (/))
2	1	necon2bi 2053	. . . 4 |- ((C i^i B) = (/) -> -. (A e. C /\ A e. B))
3		imnan 261	. . . 4 |- ((A e. C -> -. A e. B) <-> -. (A e. C /\ A e. B))
4	2, 3	sylibr 217	. . 3 |- ((C i^i B) = (/) -> (A e. C -> -. A e. B))
5	4	con2d 107	. 2 |- ((C i^i B) = (/) -> (A e. B -> -. A e. C))
6	5	impcom 378	1 |- ((A e. B /\ (C i^i B) = (/)) -> -. A e. C)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   i^i cin 2592  (/)c0 2875

This theorem is referenced by:  peano5 3975  aceq5 5902

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-in 2603  df-nul 2876

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