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Mirrors > Home > MPE Home > Th. List > Mathboxes > sltletrd | 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 ) |
sltletrd.4 | ⊢ (𝜑 → 𝐴 <s 𝐵) |
sltletrd.5 | ⊢ (𝜑 → 𝐵 ≤s 𝐶) |
Ref | Expression |
---|---|
sltletrd | ⊢ (𝜑 → 𝐴 <s 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sltletrd.4 | . 2 ⊢ (𝜑 → 𝐴 <s 𝐵) | |
2 | sltletrd.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 | sltletr 33230 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 <s 𝐵 ∧ 𝐵 ≤s 𝐶) → 𝐴 <s 𝐶)) | |
7 | 3, 4, 5, 6 | syl3anc 1367 | . 2 ⊢ (𝜑 → ((𝐴 <s 𝐵 ∧ 𝐵 ≤s 𝐶) → 𝐴 <s 𝐶)) |
8 | 1, 2, 7 | mp2and 697 | 1 ⊢ (𝜑 → 𝐴 <s 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 class class class wbr 5058 No csur 33142 <s cslt 33143 ≤s csle 33218 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-ord 6188 df-on 6189 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-1o 8096 df-2o 8097 df-no 33145 df-slt 33146 df-sle 33219 |
This theorem is referenced by: slerec 33272 |
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