Theorem minel 2328

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 2327	. . . . 5 |- ((A e. C /\ A e. B) -> (C i^i B) =/= (/))
2	1	necon2bi 1615	. . . 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 958   e. wcel 960   i^i cin 2049  (/)c0 2283

This theorem is referenced by:  peano5 3159  aceq5 4750

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-in 2054  df-nul 2284

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