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| Mirrors > Home > MPE Home > Th. List > ndmovord | Structured version Visualization version GIF version | ||
| Description: Elimination of redundant antecedents in an ordering law. (Contributed by NM, 7-Mar-1996.) | 
| 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))) | 
| Ref | Expression | 
|---|---|
| ndmovord | ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) | 
| 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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