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Mirrors > Home > MPE Home > Th. List > Mathboxes > slelttrd | 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 ) |
slelttrd.4 | ⊢ (𝜑 → 𝐴 ≤s 𝐵) |
slelttrd.5 | ⊢ (𝜑 → 𝐵 <s 𝐶) |
Ref | Expression |
---|---|
slelttrd | ⊢ (𝜑 → 𝐴 <s 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | slelttrd.4 | . 2 ⊢ (𝜑 → 𝐴 ≤s 𝐵) | |
2 | slelttrd.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 | slelttr 33593 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 ≤s 𝐵 ∧ 𝐵 <s 𝐶) → 𝐴 <s 𝐶)) | |
7 | 3, 4, 5, 6 | syl3anc 1372 | . 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 399 ∈ wcel 2113 class class class wbr 5027 No csur 33476 <s cslt 33477 ≤s csle 33580 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 ax-un 7473 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6169 df-on 6170 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-fv 6341 df-1o 8124 df-2o 8125 df-no 33479 df-slt 33480 df-sle 33581 |
This theorem is referenced by: slerec 33646 sltlpss 33717 cofsslt 33718 |
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