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Theorem ndmovordi 7549
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 5701 . . . 4 ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → ((𝐶𝐹𝐴) ∈ 𝑆 ∧ (𝐶𝐹𝐵) ∈ 𝑆))
32simpld 496 . . 3 ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → (𝐶𝐹𝐴) ∈ 𝑆)
4 ndmovordi.2 . . . . 5 dom 𝐹 = (𝑆 × 𝑆)
5 ndmovordi.5 . . . . 5 ¬ ∅ ∈ 𝑆
64, 5ndmovrcl 7544 . . . 4 ((𝐶𝐹𝐴) ∈ 𝑆 → (𝐶𝑆𝐴𝑆))
76simpld 496 . . 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 205   = wceq 1542  wcel 2107  wss 3914  c0 4286   class class class wbr 5109   × cxp 5635  dom cdm 5637  (class class class)co 7361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-xp 5643  df-dm 5647  df-iota 6452  df-fv 6508  df-ov 7364
This theorem is referenced by:  ltexprlem4  10983  ltsosr  11038
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