Theorem relsdom 4374

Description: Strict dominance is a relation.

Assertion

Ref	Expression
relsdom	|- Rel ~<

Proof of Theorem relsdom

Step	Hyp	Ref	Expression
1		reldom 4373	. 2 |- Rel ~<_
2		reldif 3264	. . 3 |- (Rel ~<_ -> Rel ( ~<_ \ ~~ ))
3		df-sdom 4370	. . . 4 |- ~< = ( ~<_ \ ~~ )
4	3	releqi 3244	. . 3 |- (Rel ~< <-> Rel ( ~<_ \ ~~ ))
5	2, 4	sylibr 200	. 2 |- (Rel ~<_ -> Rel ~< )
6	1, 5	ax-mp 7	1 |- Rel ~<

Syntax hints:   \ cdif 2044  Rel wrel 3175   ~~ cen 4364   ~<_ cdom 4365   ~< csdm 4366

This theorem is referenced by:  domnsym 4463  ensdomtr 4471  sdomirr 4472  sdomex 4473  domsdomtr 4476  alephnbtwn2 4869  alephsucdom 4880

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-11 967  ax-12 968  ax-13 969  ax-14 970  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  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-opab 2667  df-xp 3184  df-rel 3185  df-dom 4369  df-sdom 4370

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