Theorem sdomdom 4373

Description: Strict dominance implies dominance.

Assertion

Ref	Expression
sdomdom	|- (A ~< B -> A ~<_ B)

Proof of Theorem sdomdom

Step	Hyp	Ref	Expression
1		brsdom 4369	. 2 |- (A ~< B <-> (A ~<_ B /\ -. A ~~ B))
2	1	pm3.26bi 322	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:  sdomnsym 4448  sdomdomtr 4455  sdomtr 4460  isfinite2 4529  pwfi 4551  entri3 4821  sucdom 4822  sucxpdom 4826  infxpidmlem12 7514  infdif 7519  infmap1 7524  aleph1irr 7528  alephexp1 7534

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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