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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 33641 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ↔ ¬ 𝐵 <s 𝐴)) | |
2 | 1 | 3adant3 1134 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ≤s 𝐵 ↔ ¬ 𝐵 <s 𝐴)) |
3 | 2 | anbi1d 633 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 ≤s 𝐵 ∧ 𝐵 <s 𝐶) ↔ (¬ 𝐵 <s 𝐴 ∧ 𝐵 <s 𝐶))) |
4 | sltso 33565 | . . 3 ⊢ <s Or No | |
5 | sotr2 5485 | . . 3 ⊢ (( <s Or No ∧ (𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No )) → ((¬ 𝐵 <s 𝐴 ∧ 𝐵 <s 𝐶) → 𝐴 <s 𝐶)) | |
6 | 4, 5 | mpan 690 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((¬ 𝐵 <s 𝐴 ∧ 𝐵 <s 𝐶) → 𝐴 <s 𝐶)) |
7 | 3, 6 | sylbid 243 | 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 1089 ∈ wcel 2112 class class class wbr 5039 Or wor 5452 No csur 33529 <s cslt 33530 ≤s csle 33633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-ord 6194 df-on 6195 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-1o 8180 df-2o 8181 df-no 33532 df-slt 33533 df-sle 33634 |
This theorem is referenced by: slelttrd 33650 |
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