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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 27738 | . 2 ⊢ {〈1o, ∅〉, 〈1o, 2o〉, 〈∅, 2o〉} Or ({1o, 2o} ∪ {∅}) | |
2 | df-no 27705 | . 2 ⊢ No = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o}} | |
3 | df-slt 27706 | . 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 27721 | . 2 ⊢ ¬ ∅ ∈ {1o, 2o} | |
5 | 1, 2, 3, 4 | soseq 8200 | 1 ⊢ <s Or No |
Colors of variables: wff setvar class |
Syntax hints: ∅c0 4352 {cpr 4650 {ctp 4652 〈cop 4654 Or wor 5606 1oc1o 8515 2oc2o 8516 No csur 27702 <s cslt 27703 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-ord 6398 df-on 6399 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-fv 6581 df-1o 8522 df-2o 8523 df-no 27705 df-slt 27706 |
This theorem is referenced by: nosepne 27743 nosepdm 27747 nodenselem4 27750 nodenselem5 27751 nodenselem7 27753 nolt02o 27758 nogt01o 27759 noresle 27760 nomaxmo 27761 nominmo 27762 nosupprefixmo 27763 noinfprefixmo 27764 nosupbnd1lem1 27771 nosupbnd1lem2 27772 nosupbnd1lem4 27774 nosupbnd1lem6 27776 nosupbnd1 27777 nosupbnd2lem1 27778 nosupbnd2 27779 noinfbnd1lem1 27786 noinfbnd1lem2 27787 noinfbnd1lem4 27789 noinfbnd1lem6 27791 noinfbnd1 27792 noinfbnd2lem1 27793 noinfbnd2 27794 noetasuplem4 27799 noetainflem4 27803 sltirr 27809 slttr 27810 sltasym 27811 sltlin 27812 slttrieq2 27813 slttrine 27814 sleloe 27817 sltletr 27819 slelttr 27820 |
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