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Mirrors > Home > MPE Home > Th. List > Mathboxes > slelttr | Structured version Visualization version GIF version |
Description: Surreal transitive law. (Contributed by Scott Fenton, 8-Dec-2021.) |
Ref | Expression |
---|---|
slelttr | ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 ≤s 𝐵 ∧ 𝐵 <s 𝐶) → 𝐴 <s 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | slenlt 33238 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ↔ ¬ 𝐵 <s 𝐴)) | |
2 | 1 | 3adant3 1128 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ≤s 𝐵 ↔ ¬ 𝐵 <s 𝐴)) |
3 | 2 | anbi1d 631 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 ≤s 𝐵 ∧ 𝐵 <s 𝐶) ↔ (¬ 𝐵 <s 𝐴 ∧ 𝐵 <s 𝐶))) |
4 | sltso 33188 | . . 3 ⊢ <s Or No | |
5 | sotr2 5491 | . . 3 ⊢ (( <s Or No ∧ (𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No )) → ((¬ 𝐵 <s 𝐴 ∧ 𝐵 <s 𝐶) → 𝐴 <s 𝐶)) | |
6 | 4, 5 | mpan 688 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((¬ 𝐵 <s 𝐴 ∧ 𝐵 <s 𝐶) → 𝐴 <s 𝐶)) |
7 | 3, 6 | sylbid 242 | 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 208 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2114 class class class wbr 5052 Or wor 5459 No csur 33154 <s cslt 33155 ≤s csle 33230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-ord 6180 df-on 6181 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-fv 6349 df-1o 8088 df-2o 8089 df-no 33157 df-slt 33158 df-sle 33231 |
This theorem is referenced by: slelttrd 33247 |
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