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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 26976 | . . . . 5 ⊢ <s Or No | |
2 | soss 5563 | . . . . 5 ⊢ (𝑆 ⊆ No → ( <s Or No → <s Or 𝑆)) | |
3 | 1, 2 | mpi 20 | . . . 4 ⊢ (𝑆 ⊆ No → <s Or 𝑆) |
4 | cnvso 6238 | . . . 4 ⊢ ( <s Or 𝑆 ↔ ◡ <s Or 𝑆) | |
5 | 3, 4 | sylib 217 | . . 3 ⊢ (𝑆 ⊆ No → ◡ <s Or 𝑆) |
6 | somo 5580 | . . 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 3447 | . . . . . 6 ⊢ 𝑦 ∈ V | |
9 | vex 3447 | . . . . . 6 ⊢ 𝑥 ∈ V | |
10 | 8, 9 | brcnv 5836 | . . . . 5 ⊢ (𝑦◡ <s 𝑥 ↔ 𝑥 <s 𝑦) |
11 | 10 | notbii 319 | . . . 4 ⊢ (¬ 𝑦◡ <s 𝑥 ↔ ¬ 𝑥 <s 𝑦) |
12 | 11 | ralbii 3094 | . . 3 ⊢ (∀𝑦 ∈ 𝑆 ¬ 𝑦◡ <s 𝑥 ↔ ∀𝑦 ∈ 𝑆 ¬ 𝑥 <s 𝑦) |
13 | 12 | rmobii 3359 | . 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 3062 ∃*wrmo 3350 ⊆ wss 3908 class class class wbr 5103 Or wor 5542 ◡ccnv 5630 No csur 26940 <s cslt 26941 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3351 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-ord 6318 df-on 6319 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-fv 6501 df-1o 8404 df-2o 8405 df-no 26943 df-slt 26944 |
This theorem is referenced by: nosupno 27003 nosupbday 27005 nosupbnd1 27014 nosupbnd2 27016 |
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