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Mirrors > Home > MPE Home > Th. List > sltso | Structured version Visualization version GIF version |
Description: Less-than totally orders the surreals. Axiom O of [Alling] p. 184. (Contributed by Scott Fenton, 9-Jun-2011.) |
Ref | Expression |
---|---|
sltso | ⊢ <s Or No |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sltsolem1 27007 | . 2 ⊢ {⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} Or ({1o, 2o} ∪ {∅}) | |
2 | df-no 26975 | . 2 ⊢ No = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o}} | |
3 | df-slt 26976 | . 2 ⊢ <s = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ No ∧ 𝑔 ∈ No ) ∧ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑔‘𝑦) ∧ (𝑓‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔‘𝑥)))} | |
4 | nosgnn0 26990 | . 2 ⊢ ¬ ∅ ∈ {1o, 2o} | |
5 | 1, 2, 3, 4 | soseq 8087 | 1 ⊢ <s Or No |
Colors of variables: wff setvar class |
Syntax hints: ∅c0 4280 {cpr 4586 {ctp 4588 ⟨cop 4590 Or wor 5542 1oc1o 8401 2oc2o 8402 No csur 26972 <s cslt 26973 |
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 2707 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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 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 8408 df-2o 8409 df-no 26975 df-slt 26976 |
This theorem is referenced by: nosepne 27012 nosepdm 27016 nodenselem4 27019 nodenselem5 27020 nodenselem7 27022 nolt02o 27027 nogt01o 27028 noresle 27029 nomaxmo 27030 nominmo 27031 nosupprefixmo 27032 noinfprefixmo 27033 nosupbnd1lem1 27040 nosupbnd1lem2 27041 nosupbnd1lem4 27043 nosupbnd1lem6 27045 nosupbnd1 27046 nosupbnd2lem1 27047 nosupbnd2 27048 noinfbnd1lem1 27055 noinfbnd1lem2 27056 noinfbnd1lem4 27058 noinfbnd1lem6 27060 noinfbnd1 27061 noinfbnd2lem1 27062 noinfbnd2 27063 noetasuplem4 27068 noetainflem4 27072 sltirr 27078 slttr 27079 sltasym 27080 sltlin 27081 slttrieq2 27082 slttrine 27083 sleloe 27086 sltletr 27088 slelttr 27089 |
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