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Mirrors > Home > MPE Home > Th. List > Mathboxes > sletrd | Structured version Visualization version GIF version |
Description: Surreal less than or equal 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 ) |
sletrd.4 | ⊢ (𝜑 → 𝐴 ≤s 𝐵) |
sletrd.5 | ⊢ (𝜑 → 𝐵 ≤s 𝐶) |
Ref | Expression |
---|---|
sletrd | ⊢ (𝜑 → 𝐴 ≤s 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sletrd.4 | . 2 ⊢ (𝜑 → 𝐴 ≤s 𝐵) | |
2 | sletrd.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 | sletr 33134 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐶) → 𝐴 ≤s 𝐶)) | |
7 | 3, 4, 5, 6 | syl3anc 1363 | . 2 ⊢ (𝜑 → ((𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐶) → 𝐴 ≤s 𝐶)) |
8 | 1, 2, 7 | mp2and 695 | 1 ⊢ (𝜑 → 𝐴 ≤s 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 class class class wbr 5057 No csur 33044 ≤s csle 33120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-1o 8091 df-2o 8092 df-no 33047 df-slt 33048 df-sle 33121 |
This theorem is referenced by: (None) |
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