Theorem relsdom 3279

Description: Strict dominance is a relation.

Assertion

Ref	Expression
relsdom	|- Rel ~<

Proof of Theorem relsdom

Step	Hyp	Ref	Expression
1		reldom 3278	. 2 |- Rel ~<_
2		reldif 2492	. . 3 |- (Rel ~<_ -> Rel ( ~<_ \ ~~ ))
3		df-sdom 3276	. . . 4 |- ~< = ( ~<_ \ ~~ )
4		releq 2477	. . . 4 |- ( ~< = ( ~<_ \ ~~ ) -> (Rel ~< <-> Rel ( ~<_ \ ~~ )))
5	3, 4	ax-mp 6	. . 3 |- (Rel ~< <-> Rel ( ~<_ \ ~~ ))
6	2, 5	sylibr 175	. 2 |- (Rel ~<_ -> Rel ~< )
7	1, 6	ax-mp 6	1 |- Rel ~<

Syntax hints:   <-> wb 127   = wceq 1091   \ cdif 1484  Rel wrel 2415   ~~ cen 3271   ~<_ cdom 3272   ~< csdm 3273

This theorem is referenced by:  sdomirr 3314  sdomex 3315  domnsym 3365  ensdomtr 3372  domsdomtr 3374  alephnbtwn2 3675  alephsucdom 3685

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-xp 2424  df-rel 2425  df-dom 3275  df-sdom 3276

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