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Theorem ndmovord 6777
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 1264 . 2 ((𝐴𝑆𝐵𝑆) → (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))))
3 ndmovord.4 . . . . 5 𝑅 ⊆ (𝑆 × 𝑆)
43brel 5128 . . . 4 (𝐴𝑅𝐵 → (𝐴𝑆𝐵𝑆))
53brel 5128 . . . . 5 ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → ((𝐶𝐹𝐴) ∈ 𝑆 ∧ (𝐶𝐹𝐵) ∈ 𝑆))
6 ndmov.1 . . . . . . . 8 dom 𝐹 = (𝑆 × 𝑆)
7 ndmovord.5 . . . . . . . 8 ¬ ∅ ∈ 𝑆
86, 7ndmovrcl 6773 . . . . . . 7 ((𝐶𝐹𝐴) ∈ 𝑆 → (𝐶𝑆𝐴𝑆))
98simprd 479 . . . . . 6 ((𝐶𝐹𝐴) ∈ 𝑆𝐴𝑆)
106, 7ndmovrcl 6773 . . . . . . 7 ((𝐶𝐹𝐵) ∈ 𝑆 → (𝐶𝑆𝐵𝑆))
1110simprd 479 . . . . . 6 ((𝐶𝐹𝐵) ∈ 𝑆𝐵𝑆)
129, 11anim12i 589 . . . . 5 (((𝐶𝐹𝐴) ∈ 𝑆 ∧ (𝐶𝐹𝐵) ∈ 𝑆) → (𝐴𝑆𝐵𝑆))
135, 12syl 17 . . . 4 ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → (𝐴𝑆𝐵𝑆))
144, 13pm5.21ni 367 . . 3 (¬ (𝐴𝑆𝐵𝑆) → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
1514a1d 25 . 2 (¬ (𝐴𝑆𝐵𝑆) → (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))))
162, 15pm2.61i 176 1 (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wss 3555  c0 3891   class class class wbr 4613   × cxp 5072  dom cdm 5074  (class class class)co 6604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-xp 5080  df-dm 5084  df-iota 5810  df-fv 5855  df-ov 6607
This theorem is referenced by:  ltapi  9669  ltmpi  9670  ltanq  9737  ltmnq  9738  ltapr  9811  ltasr  9865
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