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Mirrors > Home > MPE Home > Th. List > Mathboxes > sltso | Structured version Visualization version GIF version |
Description: Surreal 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 33564 | . 2 ⊢ {〈1o, ∅〉, 〈1o, 2o〉, 〈∅, 2o〉} Or ({1o, 2o} ∪ {∅}) | |
2 | df-no 33532 | . 2 ⊢ No = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o}} | |
3 | df-slt 33533 | . 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 33547 | . 2 ⊢ ¬ ∅ ∈ {1o, 2o} | |
5 | 1, 2, 3, 4 | soseq 33483 | 1 ⊢ <s Or No |
Colors of variables: wff setvar class |
Syntax hints: ∅c0 4223 {cpr 4529 {ctp 4531 〈cop 4533 Or wor 5452 1oc1o 8173 2oc2o 8174 No csur 33529 <s cslt 33530 |
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 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
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 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-ord 6194 df-on 6195 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-1o 8180 df-2o 8181 df-no 33532 df-slt 33533 |
This theorem is referenced by: nosepne 33569 nosepdm 33573 nodenselem4 33576 nodenselem5 33577 nodenselem7 33579 nolt02o 33584 nogt01o 33585 noresle 33586 nomaxmo 33587 nominmo 33588 nosupprefixmo 33589 noinfprefixmo 33590 nosupbnd1lem1 33597 nosupbnd1lem2 33598 nosupbnd1lem4 33600 nosupbnd1lem6 33602 nosupbnd1 33603 nosupbnd2lem1 33604 nosupbnd2 33605 noinfbnd1lem1 33612 noinfbnd1lem2 33613 noinfbnd1lem4 33615 noinfbnd1lem6 33617 noinfbnd1 33618 noinfbnd2lem1 33619 noinfbnd2 33620 noetasuplem4 33625 noetainflem4 33629 sltirr 33635 slttr 33636 sltasym 33637 sltlin 33638 slttrieq2 33639 slttrine 33640 sleloe 33643 sltletr 33645 slelttr 33646 |
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