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Theorem ndmovord 7054
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 1143 . 2 ((𝐴𝑆𝐵𝑆) → (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))))
3 ndmovord.4 . . . . 5 𝑅 ⊆ (𝑆 × 𝑆)
43brel 5368 . . . 4 (𝐴𝑅𝐵 → (𝐴𝑆𝐵𝑆))
53brel 5368 . . . . 5 ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → ((𝐶𝐹𝐴) ∈ 𝑆 ∧ (𝐶𝐹𝐵) ∈ 𝑆))
6 ndmov.1 . . . . . . . 8 dom 𝐹 = (𝑆 × 𝑆)
7 ndmovord.5 . . . . . . . 8 ¬ ∅ ∈ 𝑆
86, 7ndmovrcl 7050 . . . . . . 7 ((𝐶𝐹𝐴) ∈ 𝑆 → (𝐶𝑆𝐴𝑆))
98simprd 485 . . . . . 6 ((𝐶𝐹𝐴) ∈ 𝑆𝐴𝑆)
106, 7ndmovrcl 7050 . . . . . . 7 ((𝐶𝐹𝐵) ∈ 𝑆 → (𝐶𝑆𝐵𝑆))
1110simprd 485 . . . . . 6 ((𝐶𝐹𝐵) ∈ 𝑆𝐵𝑆)
129, 11anim12i 602 . . . . 5 (((𝐶𝐹𝐴) ∈ 𝑆 ∧ (𝐶𝐹𝐵) ∈ 𝑆) → (𝐴𝑆𝐵𝑆))
135, 12syl 17 . . . 4 ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → (𝐴𝑆𝐵𝑆))
144, 13pm5.21ni 368 . . 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 197  wa 384  w3a 1100   = wceq 1637  wcel 2156  wss 3769  c0 4116   class class class wbr 4844   × cxp 5309  dom cdm 5311  (class class class)co 6874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-xp 5317  df-dm 5321  df-iota 6064  df-fv 6109  df-ov 6877
This theorem is referenced by:  ltapi  10010  ltmpi  10011  ltanq  10078  ltmnq  10079  ltapr  10152  ltasr  10206
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