Theorem sdomdom 4527

Description: Strict dominance implies dominance.

Assertion

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

Proof of Theorem sdomdom

Step	Hyp	Ref	Expression
1		brsdom 4522	. 2 |- (A ~< B <-> (A ~<_ B /\ -. A ~~ B))
2	1	pm3.26bi 320	1 |- (A ~< B -> A ~<_ B)

Syntax hints:  -. wn 2   -> wi 3   class class class wbr 2692   ~~ cen 4505   ~<_ cdom 4506   ~< csdm 4507

This theorem is referenced by:  sdomnsym 4607  sdomdomtr 4614  sdomtr 4619  isfinite2 4692  pwfi 4714  entri3 4990  sucdom 4992  sucxpdom 4996  infxpidmlem12 7775  infdif 7780  infmap1 7785  aleph1irr 7790  alephexp1 7796  omsubsdom 11442  omsubdom 11443  omsubel 11444

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500

This theorem depends on definitions:  df-bi 145  df-an 223  df-ex 1017  df-sb 1209  df-clab 1506  df-cleq 1511  df-clel 1514  df-v 1858  df-dif 2101  df-br 2693  df-sdom 4511

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