![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5701 | . . . 4 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → ((𝐶𝐹𝐴) ∈ 𝑆 ∧ (𝐶𝐹𝐵) ∈ 𝑆)) |
3 | 2 | simpld 496 | . . 3 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → (𝐶𝐹𝐴) ∈ 𝑆) |
4 | ndmovordi.2 | . . . . 5 ⊢ dom 𝐹 = (𝑆 × 𝑆) | |
5 | ndmovordi.5 | . . . . 5 ⊢ ¬ ∅ ∈ 𝑆 | |
6 | 4, 5 | ndmovrcl 7544 | . . . 4 ⊢ ((𝐶𝐹𝐴) ∈ 𝑆 → (𝐶 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) |
7 | 6 | simpld 496 | . . 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 248 | . 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 1542 ∈ wcel 2107 ⊆ wss 3914 ∅c0 4286 class class class wbr 5109 × cxp 5635 dom cdm 5637 (class class class)co 7361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-xp 5643 df-dm 5647 df-iota 6452 df-fv 6508 df-ov 7364 |
This theorem is referenced by: ltexprlem4 10983 ltsosr 11038 |
Copyright terms: Public domain | W3C validator |