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Theorem ndmovord 7322
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 1118 . 2 ((𝐴𝑆𝐵𝑆) → (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))))
3 ndmovord.4 . . . . 5 𝑅 ⊆ (𝑆 × 𝑆)
43brel 5585 . . . 4 (𝐴𝑅𝐵 → (𝐴𝑆𝐵𝑆))
53brel 5585 . . . . 5 ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → ((𝐶𝐹𝐴) ∈ 𝑆 ∧ (𝐶𝐹𝐵) ∈ 𝑆))
6 ndmov.1 . . . . . . . 8 dom 𝐹 = (𝑆 × 𝑆)
7 ndmovord.5 . . . . . . . 8 ¬ ∅ ∈ 𝑆
86, 7ndmovrcl 7318 . . . . . . 7 ((𝐶𝐹𝐴) ∈ 𝑆 → (𝐶𝑆𝐴𝑆))
98simprd 499 . . . . . 6 ((𝐶𝐹𝐴) ∈ 𝑆𝐴𝑆)
106, 7ndmovrcl 7318 . . . . . . 7 ((𝐶𝐹𝐵) ∈ 𝑆 → (𝐶𝑆𝐵𝑆))
1110simprd 499 . . . . . 6 ((𝐶𝐹𝐵) ∈ 𝑆𝐵𝑆)
129, 11anim12i 615 . . . . 5 (((𝐶𝐹𝐴) ∈ 𝑆 ∧ (𝐶𝐹𝐵) ∈ 𝑆) → (𝐴𝑆𝐵𝑆))
135, 12syl 17 . . . 4 ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → (𝐴𝑆𝐵𝑆))
144, 13pm5.21ni 382 . . 3 (¬ (𝐴𝑆𝐵𝑆) → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
1514a1d 25 . 2 (¬ (𝐴𝑆𝐵𝑆) → (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))))
162, 15pm2.61i 185 1 (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wss 3884  c0 4246   class class class wbr 5033   × cxp 5521  dom cdm 5523  (class class class)co 7139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-xp 5529  df-dm 5533  df-iota 6287  df-fv 6336  df-ov 7142
This theorem is referenced by:  ltapi  10318  ltmpi  10319  ltanq  10386  ltmnq  10387  ltapr  10460  ltasr  10515
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