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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 32753 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 <s 𝐵 ∧ 𝐵 <s 𝐶) → 𝐴 <s 𝐶)) | |
7 | 3, 4, 5, 6 | syl3anc 1351 | . 2 ⊢ (𝜑 → ((𝐴 <s 𝐵 ∧ 𝐵 <s 𝐶) → 𝐴 <s 𝐶)) |
8 | 1, 2, 7 | mp2and 686 | 1 ⊢ (𝜑 → 𝐴 <s 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∈ wcel 2050 class class class wbr 4929 No csur 32674 <s cslt 32675 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-ord 6032 df-on 6033 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-fv 6196 df-1o 7905 df-2o 7906 df-no 32677 df-slt 32678 |
This theorem is referenced by: conway 32791 sslttr 32795 slerec 32804 |
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