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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 5610 | . . . 4 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → ((𝐶𝐹𝐴) ∈ 𝑆 ∧ (𝐶𝐹𝐵) ∈ 𝑆)) |
3 | 2 | simpld 495 | . . 3 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → (𝐶𝐹𝐴) ∈ 𝑆) |
4 | ndmovordi.2 | . . . . 5 ⊢ dom 𝐹 = (𝑆 × 𝑆) | |
5 | ndmovordi.5 | . . . . 5 ⊢ ¬ ∅ ∈ 𝑆 | |
6 | 4, 5 | ndmovrcl 7323 | . . . 4 ⊢ ((𝐶𝐹𝐴) ∈ 𝑆 → (𝐶 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) |
7 | 6 | simpld 495 | . . 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 249 | . 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 207 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 ∅c0 4288 class class class wbr 5057 × cxp 5546 dom cdm 5548 (class class class)co 7145 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-dm 5558 df-iota 6307 df-fv 6356 df-ov 7148 |
This theorem is referenced by: ltexprlem4 10449 ltsosr 10504 |
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