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Theorem ndmovord 7624

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 1121	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))))
3		ndmovord.4	. . . . 5 ⊢ 𝑅 ⊆ (𝑆 × 𝑆)
4	3	brel 5749	. . . 4 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
5	3	brel 5749	. . . . 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 7620	. . . . . . 7 ⊢ ((𝐶𝐹𝐴) ∈ 𝑆 → (𝐶 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆))
9	8	simprd 495	. . . . . 6 ⊢ ((𝐶𝐹𝐴) ∈ 𝑆 → 𝐴 ∈ 𝑆)
10	6, 7	ndmovrcl 7620	. . . . . . 7 ⊢ ((𝐶𝐹𝐵) ∈ 𝑆 → (𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
11	10	simprd 495	. . . . . 6 ⊢ ((𝐶𝐹𝐵) ∈ 𝑆 → 𝐵 ∈ 𝑆)
12	9, 11	anim12i 613	. . . . 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 377	. . 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 182	1 ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 ∅c0 4332 class class class wbr 5142 × cxp 5682 dom cdm 5684 (class class class)co 7432

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-xp 5690 df-dm 5694 df-iota 6513 df-fv 6568 df-ov 7435

This theorem is referenced by: ltapi 10944 ltmpi 10945 ltanq 11012 ltmnq 11013 ltapr 11086 ltasr 11141

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