Theorem sdomnen 4387

Description: Strict dominance implies non-equinumerosity.

Assertion

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

Proof of Theorem sdomnen

Step	Hyp	Ref	Expression
1		brsdom 4381	. 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 2619   ~~ cen 4364   ~<_ cdom 4365   ~< csdm 4366

This theorem is referenced by:  bren2 4389  sdomnsym 4462  domnsym 4463  sdomdomtr 4469  sdomirr 4472  php5 4518  pssinfOLD 4528  isfinite2OLD 4546  pm54.43 4572  cardnn 4824  cardom 4825  ondomcard 4857  top2ind 10548

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-dif 2049  df-br 2620  df-sdom 4370

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