Theorem ndmovordi 7603

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 5724	. . . 4 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → ((𝐶𝐹𝐴) ∈ 𝑆 ∧ (𝐶𝐹𝐵) ∈ 𝑆))
3	2	simpld 494	. . 3 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → (𝐶𝐹𝐴) ∈ 𝑆)
4		ndmovordi.2	. . . . 5 ⊢ dom 𝐹 = (𝑆 × 𝑆)
5		ndmovordi.5	. . . . 5 ⊢ ¬ ∅ ∈ 𝑆
6	4, 5	ndmovrcl 7598	. . . 4 ⊢ ((𝐶𝐹𝐴) ∈ 𝑆 → (𝐶 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆))
7	6	simpld 494	. . 3 ⊢ ((𝐶𝐹𝐴) ∈ 𝑆 → 𝐶 ∈ 𝑆)
8	3, 7	syl 17	. 2 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → 𝐶 ∈ 𝑆)
9		ndmovordi.6	. . 3 ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
10	9	biimprd 248	. 2 ⊢ (𝐶 ∈ 𝑆 → ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → 𝐴𝑅𝐵))
11	8, 10	mpcom 38	1 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → 𝐴𝑅𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109   ⊆ wss 3931  ∅c0 4313   class class class wbr 5124   × cxp 5657  dom cdm 5659  (class class class)co 7410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-xp 5665  df-dm 5669  df-iota 6489  df-fv 6544  df-ov 7413

This theorem is referenced by:  ltexprlem4  11058  ltsosr  11113