MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ndmovord Structured version   Visualization version   GIF version

Theorem ndmovord 7611
Description: Elimination of redundant antecedents in an ordering law. (Contributed by NM, 7-Mar-1996.)
Hypotheses
Ref Expression
ndmov.1 dom 𝐹 = (𝑆 × 𝑆)
ndmovord.4 𝑅 ⊆ (𝑆 × 𝑆)
ndmovord.5 ¬ ∅ ∈ 𝑆
ndmovord.6 ((𝐴𝑆𝐵𝑆𝐶𝑆) → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
Assertion
Ref Expression
ndmovord (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))

Proof of Theorem ndmovord
StepHypRef Expression
1 ndmovord.6 . . 3 ((𝐴𝑆𝐵𝑆𝐶𝑆) → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
213expia 1119 . 2 ((𝐴𝑆𝐵𝑆) → (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))))
3 ndmovord.4 . . . . 5 𝑅 ⊆ (𝑆 × 𝑆)
43brel 5743 . . . 4 (𝐴𝑅𝐵 → (𝐴𝑆𝐵𝑆))
53brel 5743 . . . . 5 ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → ((𝐶𝐹𝐴) ∈ 𝑆 ∧ (𝐶𝐹𝐵) ∈ 𝑆))
6 ndmov.1 . . . . . . . 8 dom 𝐹 = (𝑆 × 𝑆)
7 ndmovord.5 . . . . . . . 8 ¬ ∅ ∈ 𝑆
86, 7ndmovrcl 7607 . . . . . . 7 ((𝐶𝐹𝐴) ∈ 𝑆 → (𝐶𝑆𝐴𝑆))
98simprd 495 . . . . . 6 ((𝐶𝐹𝐴) ∈ 𝑆𝐴𝑆)
106, 7ndmovrcl 7607 . . . . . . 7 ((𝐶𝐹𝐵) ∈ 𝑆 → (𝐶𝑆𝐵𝑆))
1110simprd 495 . . . . . 6 ((𝐶𝐹𝐵) ∈ 𝑆𝐵𝑆)
129, 11anim12i 612 . . . . 5 (((𝐶𝐹𝐴) ∈ 𝑆 ∧ (𝐶𝐹𝐵) ∈ 𝑆) → (𝐴𝑆𝐵𝑆))
135, 12syl 17 . . . 4 ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → (𝐴𝑆𝐵𝑆))
144, 13pm5.21ni 377 . . 3 (¬ (𝐴𝑆𝐵𝑆) → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
1514a1d 25 . 2 (¬ (𝐴𝑆𝐵𝑆) → (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))))
162, 15pm2.61i 182 1 (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  wss 3947  c0 4323   class class class wbr 5148   × cxp 5676  dom cdm 5678  (class class class)co 7420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5684  df-dm 5688  df-iota 6500  df-fv 6556  df-ov 7423
This theorem is referenced by:  ltapi  10926  ltmpi  10927  ltanq  10994  ltmnq  10995  ltapr  11068  ltasr  11123
  Copyright terms: Public domain W3C validator