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Mirrors > Home > MPE Home > Th. List > Mathboxes > nominmo | Structured version Visualization version GIF version |
Description: A class of surreals has at most one minimum. (Contributed by Scott Fenton, 8-Aug-2024.) |
Ref | Expression |
---|---|
nominmo | ⊢ (𝑆 ⊆ No → ∃*𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ¬ 𝑦 <s 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sltso 33476 | . . 3 ⊢ <s Or No | |
2 | soss 5466 | . . 3 ⊢ (𝑆 ⊆ No → ( <s Or No → <s Or 𝑆)) | |
3 | 1, 2 | mpi 20 | . 2 ⊢ (𝑆 ⊆ No → <s Or 𝑆) |
4 | somo 5483 | . 2 ⊢ ( <s Or 𝑆 → ∃*𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ¬ 𝑦 <s 𝑥) | |
5 | 3, 4 | syl 17 | 1 ⊢ (𝑆 ⊆ No → ∃*𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ¬ 𝑦 <s 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wral 3070 ∃*wrmo 3073 ⊆ wss 3860 class class class wbr 5036 Or wor 5446 No csur 33440 <s cslt 33441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-sep 5173 ax-nul 5180 ax-pr 5302 ax-un 7465 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-ral 3075 df-rex 3076 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-br 5037 df-opab 5099 df-mpt 5117 df-tr 5143 df-id 5434 df-eprel 5439 df-po 5447 df-so 5448 df-fr 5487 df-we 5489 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-ord 6177 df-on 6178 df-suc 6180 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-fv 6348 df-1o 8118 df-2o 8119 df-no 33443 df-slt 33444 |
This theorem is referenced by: noinfno 33518 noinfbday 33520 noinfbnd1 33529 noinfbnd2 33531 |
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