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Mirrors > Home > MPE Home > Th. List > nomaxmo | Structured version Visualization version GIF version |
Description: A class of surreals has at most one maximum. (Contributed by Scott Fenton, 5-Dec-2021.) |
Ref | Expression |
---|---|
nomaxmo | ⊢ (𝑆 ⊆ No → ∃*𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ¬ 𝑥 <s 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sltso 27047 | . . . . 5 ⊢ <s Or No | |
2 | soss 5569 | . . . . 5 ⊢ (𝑆 ⊆ No → ( <s Or No → <s Or 𝑆)) | |
3 | 1, 2 | mpi 20 | . . . 4 ⊢ (𝑆 ⊆ No → <s Or 𝑆) |
4 | cnvso 6244 | . . . 4 ⊢ ( <s Or 𝑆 ↔ ◡ <s Or 𝑆) | |
5 | 3, 4 | sylib 217 | . . 3 ⊢ (𝑆 ⊆ No → ◡ <s Or 𝑆) |
6 | somo 5586 | . . 3 ⊢ (◡ <s Or 𝑆 → ∃*𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ¬ 𝑦◡ <s 𝑥) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝑆 ⊆ No → ∃*𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ¬ 𝑦◡ <s 𝑥) |
8 | vex 3451 | . . . . . 6 ⊢ 𝑦 ∈ V | |
9 | vex 3451 | . . . . . 6 ⊢ 𝑥 ∈ V | |
10 | 8, 9 | brcnv 5842 | . . . . 5 ⊢ (𝑦◡ <s 𝑥 ↔ 𝑥 <s 𝑦) |
11 | 10 | notbii 320 | . . . 4 ⊢ (¬ 𝑦◡ <s 𝑥 ↔ ¬ 𝑥 <s 𝑦) |
12 | 11 | ralbii 3093 | . . 3 ⊢ (∀𝑦 ∈ 𝑆 ¬ 𝑦◡ <s 𝑥 ↔ ∀𝑦 ∈ 𝑆 ¬ 𝑥 <s 𝑦) |
13 | 12 | rmobii 3360 | . 2 ⊢ (∃*𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ¬ 𝑦◡ <s 𝑥 ↔ ∃*𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ¬ 𝑥 <s 𝑦) |
14 | 7, 13 | sylib 217 | 1 ⊢ (𝑆 ⊆ No → ∃*𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ¬ 𝑥 <s 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wral 3061 ∃*wrmo 3351 ⊆ wss 3914 class class class wbr 5109 Or wor 5548 ◡ccnv 5636 No csur 27011 <s cslt 27012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-ord 6324 df-on 6325 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fv 6508 df-1o 8416 df-2o 8417 df-no 27014 df-slt 27015 |
This theorem is referenced by: nosupno 27074 nosupbday 27076 nosupbnd1 27085 nosupbnd2 27087 |
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