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Theorem ndmovordi 7595
Description: Elimination of redundant antecedent in an ordering law. (Contributed by NM, 25-Jun-1998.)
Hypotheses
Ref Expression
ndmovordi.2 dom 𝐹 = (𝑆 × 𝑆)
ndmovordi.4 𝑅 ⊆ (𝑆 × 𝑆)
ndmovordi.5 ¬ ∅ ∈ 𝑆
ndmovordi.6 (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
Assertion
Ref Expression
ndmovordi ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → 𝐴𝑅𝐵)

Proof of Theorem ndmovordi
StepHypRef Expression
1 ndmovordi.4 . . . . 5 𝑅 ⊆ (𝑆 × 𝑆)
21brel 5734 . . . 4 ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → ((𝐶𝐹𝐴) ∈ 𝑆 ∧ (𝐶𝐹𝐵) ∈ 𝑆))
32simpld 494 . . 3 ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → (𝐶𝐹𝐴) ∈ 𝑆)
4 ndmovordi.2 . . . . 5 dom 𝐹 = (𝑆 × 𝑆)
5 ndmovordi.5 . . . . 5 ¬ ∅ ∈ 𝑆
64, 5ndmovrcl 7590 . . . 4 ((𝐶𝐹𝐴) ∈ 𝑆 → (𝐶𝑆𝐴𝑆))
76simpld 494 . . 3 ((𝐶𝐹𝐴) ∈ 𝑆𝐶𝑆)
83, 7syl 17 . 2 ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → 𝐶𝑆)
9 ndmovordi.6 . . 3 (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
109biimprd 247 . 2 (𝐶𝑆 → ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → 𝐴𝑅𝐵))
118, 10mpcom 38 1 ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → 𝐴𝑅𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1533  wcel 2098  wss 3943  c0 4317   class class class wbr 5141   × cxp 5667  dom cdm 5669  (class class class)co 7405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-dm 5679  df-iota 6489  df-fv 6545  df-ov 7408
This theorem is referenced by:  ltexprlem4  11036  ltsosr  11091
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