Theorem sdomnen 4374

Description: Strict dominance implies non-equinumerosity.

Assertion

Ref	Expression
sdomnen	|- (A ~< B -> -. A ~~ B)

Proof of Theorem sdomnen

Step	Hyp	Ref	Expression
1		brsdom 4369	. 2 |- (A ~< B <-> (A ~<_ B /\ -. A ~~ B))
2	1	pm3.27bi 326	1 |- (A ~< B -> -. A ~~ B)

Syntax hints:  -. wn 2   -> wi 3   class class class wbr 2614   ~~ cen 4354   ~<_ cdom 4355   ~< csdm 4356

This theorem is referenced by:  bren2 4376  sdomnsym 4448  domnsym 4449  sdomdomtr 4455  sdomirr 4458  php5 4503  pssinf 4513  isfinite2 4529  pm54.43 4552  cardnn 4804  cardom 4805  ondomcard 4837

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-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-dif 2045  df-br 2615  df-sdom 4359

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