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Theorem ndmovordi 7328
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 5610 . . . 4 ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → ((𝐶𝐹𝐴) ∈ 𝑆 ∧ (𝐶𝐹𝐵) ∈ 𝑆))
32simpld 495 . . 3 ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → (𝐶𝐹𝐴) ∈ 𝑆)
4 ndmovordi.2 . . . . 5 dom 𝐹 = (𝑆 × 𝑆)
5 ndmovordi.5 . . . . 5 ¬ ∅ ∈ 𝑆
64, 5ndmovrcl 7323 . . . 4 ((𝐶𝐹𝐴) ∈ 𝑆 → (𝐶𝑆𝐴𝑆))
76simpld 495 . . 3 ((𝐶𝐹𝐴) ∈ 𝑆𝐶𝑆)
83, 7syl 17 . 2 ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → 𝐶𝑆)
9 ndmovordi.6 . . 3 (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
109biimprd 249 . 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 207   = wceq 1528  wcel 2105  wss 3933  c0 4288   class class class wbr 5057   × cxp 5546  dom cdm 5548  (class class class)co 7145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-dm 5558  df-iota 6307  df-fv 6356  df-ov 7148
This theorem is referenced by:  ltexprlem4  10449  ltsosr  10504
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