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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 1117 | . 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))) |
3 | ndmovord.4 | . . . . 5 ⊢ 𝑅 ⊆ (𝑆 × 𝑆) | |
4 | 3 | brel 5619 | . . . 4 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) |
5 | 3 | brel 5619 | . . . . 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 7336 | . . . . . . 7 ⊢ ((𝐶𝐹𝐴) ∈ 𝑆 → (𝐶 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) |
9 | 8 | simprd 498 | . . . . . 6 ⊢ ((𝐶𝐹𝐴) ∈ 𝑆 → 𝐴 ∈ 𝑆) |
10 | 6, 7 | ndmovrcl 7336 | . . . . . . 7 ⊢ ((𝐶𝐹𝐵) ∈ 𝑆 → (𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) |
11 | 10 | simprd 498 | . . . . . 6 ⊢ ((𝐶𝐹𝐵) ∈ 𝑆 → 𝐵 ∈ 𝑆) |
12 | 9, 11 | anim12i 614 | . . . . 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 381 | . . 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 184 | 1 ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 ∅c0 4293 class class class wbr 5068 × cxp 5555 dom cdm 5557 (class class class)co 7158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-xp 5563 df-dm 5567 df-iota 6316 df-fv 6365 df-ov 7161 |
This theorem is referenced by: ltapi 10327 ltmpi 10328 ltanq 10395 ltmnq 10396 ltapr 10469 ltasr 10524 |
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