![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 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 27056 | . . . . . . 7 ⊢ ((𝐶 ∈ No ∧ 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐶 <s 𝐴 ∧ 𝐴 ≤s 𝐵) → 𝐶 <s 𝐵)) | |
2 | 1 | 3coml 1127 | . . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐶 <s 𝐴 ∧ 𝐴 ≤s 𝐵) → 𝐶 <s 𝐵)) |
3 | 2 | expcomd 417 | . . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ≤s 𝐵 → (𝐶 <s 𝐴 → 𝐶 <s 𝐵))) |
4 | 3 | imp 407 | . . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐴 ≤s 𝐵) → (𝐶 <s 𝐴 → 𝐶 <s 𝐵)) |
5 | 4 | con3d 152 | . . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐴 ≤s 𝐵) → (¬ 𝐶 <s 𝐵 → ¬ 𝐶 <s 𝐴)) |
6 | 5 | expimpd 454 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 ≤s 𝐵 ∧ ¬ 𝐶 <s 𝐵) → ¬ 𝐶 <s 𝐴)) |
7 | slenlt 27052 | . . . 4 ⊢ ((𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 ≤s 𝐶 ↔ ¬ 𝐶 <s 𝐵)) | |
8 | 7 | 3adant1 1130 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 ≤s 𝐶 ↔ ¬ 𝐶 <s 𝐵)) |
9 | 8 | anbi2d 629 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐶) ↔ (𝐴 ≤s 𝐵 ∧ ¬ 𝐶 <s 𝐵))) |
10 | slenlt 27052 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ≤s 𝐶 ↔ ¬ 𝐶 <s 𝐴)) | |
11 | 10 | 3adant2 1131 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ≤s 𝐶 ↔ ¬ 𝐶 <s 𝐴)) |
12 | 6, 9, 11 | 3imtr4d 293 | 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 205 ∧ wa 396 ∧ w3a 1087 ∈ wcel 2106 class class class wbr 5103 No csur 26940 <s cslt 26941 ≤s csle 27044 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-ord 6318 df-on 6319 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-fv 6501 df-1o 8404 df-2o 8405 df-no 26943 df-slt 26944 df-sle 27045 |
This theorem is referenced by: sletrd 27062 |
Copyright terms: Public domain | W3C validator |