Theorem ndmovordi 7612

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 5715	. . . 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 7607	. . . 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 1569   ∈ wcel 2144   ⊆ wss 3895  ∅c0 4278   class class class wbr 5102   × cxp 5648  dom cdm 5650  (class class class)co 7420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5656  df-dm 5660  df-iota 6494  df-fv 6546  df-ov 7423

This theorem is used by:  ltexprlem4  11124  ltsosr  11179