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Mirrors > Home > MPE Home > Th. List > ndmovordi | Structured version Visualization version GIF version |
Description: Elimination of redundant antecedent in an ordering law. (Contributed by NM, 25-Jun-1998.) |
Ref | Expression |
---|---|
ndmovordi.2 | ⊢ dom 𝐹 = (𝑆 × 𝑆) |
ndmovordi.4 | ⊢ 𝑅 ⊆ (𝑆 × 𝑆) |
ndmovordi.5 | ⊢ ¬ ∅ ∈ 𝑆 |
ndmovordi.6 | ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) |
Ref | Expression |
---|---|
ndmovordi | ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → 𝐴𝑅𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ndmovordi.4 | . . . . 5 ⊢ 𝑅 ⊆ (𝑆 × 𝑆) | |
2 | 1 | brel 5306 | . . . 4 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → ((𝐶𝐹𝐴) ∈ 𝑆 ∧ (𝐶𝐹𝐵) ∈ 𝑆)) |
3 | 2 | simpld 482 | . . 3 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → (𝐶𝐹𝐴) ∈ 𝑆) |
4 | ndmovordi.2 | . . . . 5 ⊢ dom 𝐹 = (𝑆 × 𝑆) | |
5 | ndmovordi.5 | . . . . 5 ⊢ ¬ ∅ ∈ 𝑆 | |
6 | 4, 5 | ndmovrcl 6965 | . . . 4 ⊢ ((𝐶𝐹𝐴) ∈ 𝑆 → (𝐶 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) |
7 | 6 | simpld 482 | . . 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 238 | . 2 ⊢ (𝐶 ∈ 𝑆 → ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → 𝐴𝑅𝐵)) |
11 | 8, 10 | mpcom 38 | 1 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → 𝐴𝑅𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 = wceq 1631 ∈ wcel 2145 ⊆ wss 3723 ∅c0 4063 class class class wbr 4786 × cxp 5247 dom cdm 5249 (class class class)co 6791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-xp 5255 df-dm 5259 df-iota 5992 df-fv 6037 df-ov 6794 |
This theorem is referenced by: ltexprlem4 10061 ltsosr 10115 |
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