Theorem relsdom 8575

Description: Strict dominance is a relation. (Contributed by NM, 31-Mar-1998.)

Assertion

Ref	Expression
relsdom	⊢ Rel ≺

Proof of Theorem relsdom

Step	Hyp	Ref	Expression
1		reldom 8574	. 2 ⊢ Rel ≼
2		reldif 5669	. . 3 ⊢ (Rel ≼ → Rel ( ≼ ∖ ≈ ))
3		df-sdom 8571	. . . 4 ⊢ ≺ = ( ≼ ∖ ≈ )
4	3	releqi 5633	. . 3 ⊢ (Rel ≺ ↔ Rel ( ≼ ∖ ≈ ))
5	2, 4	sylibr 237	. 2 ⊢ (Rel ≼ → Rel ≺ )
6	1, 5	ax-mp 5	1 ⊢ Rel ≺

Syntax hints:   ∖ cdif 3850  Rel wrel 5540   ≈ cen 8565   ≼ cdom 8566   ≺ csdm 8567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-v 3402  df-dif 3856  df-in 3860  df-ss 3870  df-opab 5103  df-xp 5541  df-rel 5542  df-dom 8570  df-sdom 8571

This theorem is referenced by:  domdifsn  8662  sucdom2  8689  sdom0  8712  sdomirr  8717  sdomdif  8728  sdom1  8810  unxpdom  8817  unxpdom2  8818  sucxpdom  8819  isfinite2  8863  fin2inf  8868  card2on  9104  djuxpdom  9698  djufi  9699  infdif  9722  cfslb2n  9781  isfin5  9812  isfin6  9813  isfin4p1  9828  fin56  9906  fin67  9908  sdomsdomcard  10073  gchi  10137  canthp1lem1  10165  canthp1lem2  10166  canthp1  10167  frgpnabl  19127  fphpd  40251