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Theorem ndmovordi 7625
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 5749 . . . 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 7620 . . . 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 248 . 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 206   = wceq 1539  wcel 2107  wss 3950  c0 4332   class class class wbr 5142   × cxp 5682  dom cdm 5684  (class class class)co 7432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-xp 5690  df-dm 5694  df-iota 6513  df-fv 6568  df-ov 7435
This theorem is referenced by:  ltexprlem4  11080  ltsosr  11135
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