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Theorem ndmovord 6722

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	⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))

**Assertion**
Ref	Expression
ndmovord	⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))

**Proof of Theorem ndmovord**
Step	Hyp	Ref	Expression
1		ndmovord.6	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
2	1	3expia 1259	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))))
3		ndmovord.4	. . . . 5 ⊢ 𝑅 ⊆ (𝑆 × 𝑆)
4	3	brel 5090	. . . 4 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
5	3	brel 5090	. . . . 5 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → ((𝐶𝐹𝐴) ∈ 𝑆 ∧ (𝐶𝐹𝐵) ∈ 𝑆))
6		ndmov.1	. . . . . . . 8 ⊢ dom 𝐹 = (𝑆 × 𝑆)
7		ndmovord.5	. . . . . . . 8 ⊢ ¬ ∅ ∈ 𝑆
8	6, 7	ndmovrcl 6718	. . . . . . 7 ⊢ ((𝐶𝐹𝐴) ∈ 𝑆 → (𝐶 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆))
9	8	simprd 478	. . . . . 6 ⊢ ((𝐶𝐹𝐴) ∈ 𝑆 → 𝐴 ∈ 𝑆)
10	6, 7	ndmovrcl 6718	. . . . . . 7 ⊢ ((𝐶𝐹𝐵) ∈ 𝑆 → (𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
11	10	simprd 478	. . . . . 6 ⊢ ((𝐶𝐹𝐵) ∈ 𝑆 → 𝐵 ∈ 𝑆)
12	9, 11	anim12i 588	. . . . 5 ⊢ (((𝐶𝐹𝐴) ∈ 𝑆 ∧ (𝐶𝐹𝐵) ∈ 𝑆) → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
13	5, 12	syl 17	. . . 4 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
14	4, 13	pm5.21ni 366	. . 3 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
15	14	a1d 25	. 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))))
16	2, 15	pm2.61i 175	1 ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ∅c0 3874 class class class wbr 4583 × cxp 5036 dom cdm 5038 (class class class)co 6549

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-dm 5048 df-iota 5768 df-fv 5812 df-ov 6552

This theorem is referenced by: ltapi 9604 ltmpi 9605 ltanq 9672 ltmnq 9673 ltapr 9746 ltasr 9800

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