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Mirrors > Home > MPE Home > Th. List > Mathboxes > sletr | Structured version Visualization version GIF version |
Description: Surreal transitive law. (Contributed by Scott Fenton, 8-Dec-2021.) |
Ref | Expression |
---|---|
sletr | ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐶) → 𝐴 ≤s 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sltletr 33348 | . . . . . . 7 ⊢ ((𝐶 ∈ No ∧ 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐶 <s 𝐴 ∧ 𝐴 ≤s 𝐵) → 𝐶 <s 𝐵)) | |
2 | 1 | 3coml 1124 | . . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐶 <s 𝐴 ∧ 𝐴 ≤s 𝐵) → 𝐶 <s 𝐵)) |
3 | 2 | expcomd 420 | . . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ≤s 𝐵 → (𝐶 <s 𝐴 → 𝐶 <s 𝐵))) |
4 | 3 | imp 410 | . . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐴 ≤s 𝐵) → (𝐶 <s 𝐴 → 𝐶 <s 𝐵)) |
5 | 4 | con3d 155 | . . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐴 ≤s 𝐵) → (¬ 𝐶 <s 𝐵 → ¬ 𝐶 <s 𝐴)) |
6 | 5 | expimpd 457 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 ≤s 𝐵 ∧ ¬ 𝐶 <s 𝐵) → ¬ 𝐶 <s 𝐴)) |
7 | slenlt 33344 | . . . 4 ⊢ ((𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 ≤s 𝐶 ↔ ¬ 𝐶 <s 𝐵)) | |
8 | 7 | 3adant1 1127 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 ≤s 𝐶 ↔ ¬ 𝐶 <s 𝐵)) |
9 | 8 | anbi2d 631 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐶) ↔ (𝐴 ≤s 𝐵 ∧ ¬ 𝐶 <s 𝐵))) |
10 | slenlt 33344 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ≤s 𝐶 ↔ ¬ 𝐶 <s 𝐴)) | |
11 | 10 | 3adant2 1128 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ≤s 𝐶 ↔ ¬ 𝐶 <s 𝐴)) |
12 | 6, 9, 11 | 3imtr4d 297 | 1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐶) → 𝐴 ≤s 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 ∈ wcel 2111 class class class wbr 5030 No csur 33260 <s cslt 33261 ≤s csle 33336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6162 df-on 6163 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-1o 8085 df-2o 8086 df-no 33263 df-slt 33264 df-sle 33337 |
This theorem is referenced by: sletrd 33354 |
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