Theorem relsdom 4356

Description: Strict dominance is a relation.

Assertion

Ref	Expression
relsdom	|- Rel ~<

Proof of Theorem relsdom

Step	Hyp	Ref	Expression
1		reldom 4355	. 2 |- Rel ~<_
2		reldif 3254	. . 3 |- (Rel ~<_ -> Rel ( ~<_ \ ~~ ))
3		df-sdom 4353	. . . 4 |- ~< = ( ~<_ \ ~~ )
4	3	releqi 3234	. . 3 |- (Rel ~< <-> Rel ( ~<_ \ ~~ ))
5	2, 4	sylibr 200	. 2 |- (Rel ~<_ -> Rel ~< )
6	1, 5	ax-mp 7	1 |- Rel ~<

Syntax hints:   \ cdif 2034  Rel wrel 3165   ~~ cen 4348   ~<_ cdom 4349   ~< csdm 4350

This theorem is referenced by:  domnsym 4443  ensdomtr 4451  sdomirr 4452  sdomex 4453  domsdomtr 4456  alephnbtwn2 4841  alephsucdom 4852

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-opab 2657  df-xp 3174  df-rel 3175  df-dom 4352  df-sdom 4353

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