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Mirrors > Home > MPE Home > Th. List > slttrd | Structured version Visualization version GIF version |
Description: Surreal less-than is transitive. (Contributed by Scott Fenton, 8-Dec-2021.) |
Ref | Expression |
---|---|
slttrd.1 | ⊢ (𝜑 → 𝐴 ∈ No ) |
slttrd.2 | ⊢ (𝜑 → 𝐵 ∈ No ) |
slttrd.3 | ⊢ (𝜑 → 𝐶 ∈ No ) |
slttrd.4 | ⊢ (𝜑 → 𝐴 <s 𝐵) |
slttrd.5 | ⊢ (𝜑 → 𝐵 <s 𝐶) |
Ref | Expression |
---|---|
slttrd | ⊢ (𝜑 → 𝐴 <s 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | slttrd.4 | . 2 ⊢ (𝜑 → 𝐴 <s 𝐵) | |
2 | slttrd.5 | . 2 ⊢ (𝜑 → 𝐵 <s 𝐶) | |
3 | slttrd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ No ) | |
4 | slttrd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ No ) | |
5 | slttrd.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ No ) | |
6 | slttr 27807 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 <s 𝐵 ∧ 𝐵 <s 𝐶) → 𝐴 <s 𝐶)) | |
7 | 3, 4, 5, 6 | syl3anc 1370 | . 2 ⊢ (𝜑 → ((𝐴 <s 𝐵 ∧ 𝐵 <s 𝐶) → 𝐴 <s 𝐶)) |
8 | 1, 2, 7 | mp2and 699 | 1 ⊢ (𝜑 → 𝐴 <s 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 class class class wbr 5148 No csur 27699 <s cslt 27700 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-1o 8505 df-2o 8506 df-no 27702 df-slt 27703 |
This theorem is referenced by: conway 27859 sslttr 27867 slerec 27879 sltlpss 27960 cofcutr 27973 addsproplem2 28018 addsproplem6 28022 slt2addd 28061 negsproplem6 28080 mulsproplem5 28161 mulsproplem6 28162 mulsproplem7 28163 mulsproplem8 28164 mulsproplem13 28169 mulsproplem14 28170 precsexlem8 28253 precsexlem9 28254 precsexlem11 28256 om2noseqlt 28320 zscut 28408 recut 28443 |
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