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Theorem ndmovord 7154
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 1101 . 2 ((𝐴𝑆𝐵𝑆) → (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))))
3 ndmovord.4 . . . . 5 𝑅 ⊆ (𝑆 × 𝑆)
43brel 5467 . . . 4 (𝐴𝑅𝐵 → (𝐴𝑆𝐵𝑆))
53brel 5467 . . . . 5 ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → ((𝐶𝐹𝐴) ∈ 𝑆 ∧ (𝐶𝐹𝐵) ∈ 𝑆))
6 ndmov.1 . . . . . . . 8 dom 𝐹 = (𝑆 × 𝑆)
7 ndmovord.5 . . . . . . . 8 ¬ ∅ ∈ 𝑆
86, 7ndmovrcl 7150 . . . . . . 7 ((𝐶𝐹𝐴) ∈ 𝑆 → (𝐶𝑆𝐴𝑆))
98simprd 488 . . . . . 6 ((𝐶𝐹𝐴) ∈ 𝑆𝐴𝑆)
106, 7ndmovrcl 7150 . . . . . . 7 ((𝐶𝐹𝐵) ∈ 𝑆 → (𝐶𝑆𝐵𝑆))
1110simprd 488 . . . . . 6 ((𝐶𝐹𝐵) ∈ 𝑆𝐵𝑆)
129, 11anim12i 603 . . . . 5 (((𝐶𝐹𝐴) ∈ 𝑆 ∧ (𝐶𝐹𝐵) ∈ 𝑆) → (𝐴𝑆𝐵𝑆))
135, 12syl 17 . . . 4 ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → (𝐴𝑆𝐵𝑆))
144, 13pm5.21ni 370 . . 3 (¬ (𝐴𝑆𝐵𝑆) → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
1514a1d 25 . 2 (¬ (𝐴𝑆𝐵𝑆) → (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))))
162, 15pm2.61i 177 1 (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  w3a 1068   = wceq 1507  wcel 2050  wss 3830  c0 4179   class class class wbr 4929   × cxp 5405  dom cdm 5407  (class class class)co 6976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-xp 5413  df-dm 5417  df-iota 6152  df-fv 6196  df-ov 6979
This theorem is referenced by:  ltapi  10123  ltmpi  10124  ltanq  10191  ltmnq  10192  ltapr  10265  ltasr  10320
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