Theorem sdomnen 3291

Description: Strict dominance implies non-equinumerosity.

Assertion

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

Proof of Theorem sdomnen

Step	Hyp	Ref	Expression
1		brsdom 3286	. 2 |- (A ~< B <-> (A ~<_ B /\ -. A ~~ B))
2	1	pm3.27bd 263	1 |- (A ~< B -> -. A ~~ B)

Syntax hints:  -. wn 1   -> wi 2   class class class wbr 2054   ~~ cen 3271   ~<_ cdom 3272   ~< csdm 3273

This theorem is referenced by:  bren2 3293  sdomirr 3314  sdomnsym 3364  domnsym 3365  sdomdomtr 3370  php5 3413  pssinf 3422  isfinite2 3437  cardnn 3631  cardom 3632  ondomcard 3663

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-br 2063  df-sdom 3276

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