Theorem ndmovordi 7636

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	⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))

Assertion

Ref	Expression
ndmovordi	⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → 𝐴𝑅𝐵)

Proof of Theorem ndmovordi

Step	Hyp	Ref	Expression
1		ndmovordi.4	. . . . 5 ⊢ 𝑅 ⊆ (𝑆 × 𝑆)
2	1	brel 5720	. . . 4 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → ((𝐶𝐹𝐴) ∈ 𝑆 ∧ (𝐶𝐹𝐵) ∈ 𝑆))
3	2	simpld 500	. . 3 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → (𝐶𝐹𝐴) ∈ 𝑆)
4		ndmovordi.2	. . . . 5 ⊢ dom 𝐹 = (𝑆 × 𝑆)
5		ndmovordi.5	. . . . 5 ⊢ ¬ ∅ ∈ 𝑆
6	4, 5	ndmovrcl 7631	. . . 4 ⊢ ((𝐶𝐹𝐴) ∈ 𝑆 → (𝐶 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆))
7	6	simpld 500	. . 3 ⊢ ((𝐶𝐹𝐴) ∈ 𝑆 → 𝐶 ∈ 𝑆)
8	3, 7	syl 18	. 2 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → 𝐶 ∈ 𝑆)
9		ndmovordi.6	. . 3 ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
10	9	biimprd 251	. 2 ⊢ (𝐶 ∈ 𝑆 → ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → 𝐴𝑅𝐵))
11	8, 10	mpcom 39	1 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → 𝐴𝑅𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900  ∅c0 4281   class class class wbr 5106   × cxp 5653  dom cdm 5655  (class class class)co 7443

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-xp 5661  df-dm 5665  df-iota 6512  df-fv 6566  df-ov 7446

This theorem is used by:  ltexprlem4  11151  ltsosr  11206