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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 5747 | . . . 4 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → ((𝐶𝐹𝐴) ∈ 𝑆 ∧ (𝐶𝐹𝐵) ∈ 𝑆)) |
3 | 2 | simpld 493 | . . 3 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → (𝐶𝐹𝐴) ∈ 𝑆) |
4 | ndmovordi.2 | . . . . 5 ⊢ dom 𝐹 = (𝑆 × 𝑆) | |
5 | ndmovordi.5 | . . . . 5 ⊢ ¬ ∅ ∈ 𝑆 | |
6 | 4, 5 | ndmovrcl 7614 | . . . 4 ⊢ ((𝐶𝐹𝐴) ∈ 𝑆 → (𝐶 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) |
7 | 6 | simpld 493 | . . 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 247 | . 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 205 = wceq 1533 ∈ wcel 2098 ⊆ wss 3949 ∅c0 4326 class class class wbr 5152 × cxp 5680 dom cdm 5682 (class class class)co 7426 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-xp 5688 df-dm 5692 df-iota 6505 df-fv 6561 df-ov 7429 |
This theorem is referenced by: ltexprlem4 11072 ltsosr 11127 |
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