![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 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 27736 | . . 3 ⊢ <s Or No | |
2 | soss 5617 | . . 3 ⊢ (𝑆 ⊆ No → ( <s Or No → <s Or 𝑆)) | |
3 | 1, 2 | mpi 20 | . 2 ⊢ (𝑆 ⊆ No → <s Or 𝑆) |
4 | somo 5635 | . 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 3059 ∃*wrmo 3377 ⊆ wss 3963 class class class wbr 5148 Or wor 5596 No csur 27699 <s cslt 27700 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-1o 8505 df-2o 8506 df-no 27702 df-slt 27703 |
This theorem is referenced by: noinfno 27778 noinfbday 27780 noinfbnd1 27789 noinfbnd2 27791 |
Copyright terms: Public domain | W3C validator |