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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 5620 | . . . 4 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → ((𝐶𝐹𝐴) ∈ 𝑆 ∧ (𝐶𝐹𝐵) ∈ 𝑆)) |
3 | 2 | simpld 497 | . . 3 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → (𝐶𝐹𝐴) ∈ 𝑆) |
4 | ndmovordi.2 | . . . . 5 ⊢ dom 𝐹 = (𝑆 × 𝑆) | |
5 | ndmovordi.5 | . . . . 5 ⊢ ¬ ∅ ∈ 𝑆 | |
6 | 4, 5 | ndmovrcl 7337 | . . . 4 ⊢ ((𝐶𝐹𝐴) ∈ 𝑆 → (𝐶 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) |
7 | 6 | simpld 497 | . . 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 250 | . 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 208 = wceq 1536 ∈ wcel 2113 ⊆ wss 3939 ∅c0 4294 class class class wbr 5069 × cxp 5556 dom cdm 5558 (class class class)co 7159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-xp 5564 df-dm 5568 df-iota 6317 df-fv 6366 df-ov 7162 |
This theorem is referenced by: ltexprlem4 10464 ltsosr 10519 |
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