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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 33616 | . . . . 5 ⊢ <s Or No | |
2 | soss 5488 | . . . . 5 ⊢ (𝑆 ⊆ No → ( <s Or No → <s Or 𝑆)) | |
3 | 1, 2 | mpi 20 | . . . 4 ⊢ (𝑆 ⊆ No → <s Or 𝑆) |
4 | cnvso 6151 | . . . 4 ⊢ ( <s Or 𝑆 ↔ ◡ <s Or 𝑆) | |
5 | 3, 4 | sylib 221 | . . 3 ⊢ (𝑆 ⊆ No → ◡ <s Or 𝑆) |
6 | somo 5505 | . . 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 3412 | . . . . . 6 ⊢ 𝑦 ∈ V | |
9 | vex 3412 | . . . . . 6 ⊢ 𝑥 ∈ V | |
10 | 8, 9 | brcnv 5751 | . . . . 5 ⊢ (𝑦◡ <s 𝑥 ↔ 𝑥 <s 𝑦) |
11 | 10 | notbii 323 | . . . 4 ⊢ (¬ 𝑦◡ <s 𝑥 ↔ ¬ 𝑥 <s 𝑦) |
12 | 11 | ralbii 3088 | . . 3 ⊢ (∀𝑦 ∈ 𝑆 ¬ 𝑦◡ <s 𝑥 ↔ ∀𝑦 ∈ 𝑆 ¬ 𝑥 <s 𝑦) |
13 | 12 | rmobii 3308 | . 2 ⊢ (∃*𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ¬ 𝑦◡ <s 𝑥 ↔ ∃*𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ¬ 𝑥 <s 𝑦) |
14 | 7, 13 | sylib 221 | 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 3064 ⊆ wss 3866 class class class wbr 5053 Or wor 5467 ◡ccnv 5550 No csur 33580 <s cslt 33581 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-ord 6216 df-on 6217 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-fv 6388 df-1o 8202 df-2o 8203 df-no 33583 df-slt 33584 |
This theorem is referenced by: nosupno 33643 nosupbday 33645 nosupbnd1 33654 nosupbnd2 33656 |
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