Theorem sdomdom 4386

Description: Strict dominance implies dominance.

Assertion

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

Proof of Theorem sdomdom

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

This theorem is referenced by:  sdomnsym 4462  sdomdomtr 4469  sdomtr 4474  isfinite2OLD 4546  pwfiOLD 4571  entri3 4841  sucdom 4842  sucxpdom 4846  infxpidmlem12 7563  infdif 7568  infmap1 7573  aleph1irr 7578  alephexp1 7584

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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