Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > sltso | Structured version Visualization version GIF version |
Description: Surreal less than totally orders the surreals. Alling's axiom (O). (Contributed by Scott Fenton, 9-Jun-2011.) |
Ref | Expression |
---|---|
sltso | ⊢ <s Or No |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sltsolem1 31826 | . 2 ⊢ {〈1𝑜, ∅〉, 〈1𝑜, 2𝑜〉, 〈∅, 2𝑜〉} Or ({1𝑜, 2𝑜} ∪ {∅}) | |
2 | df-no 31796 | . 2 ⊢ No = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶{1𝑜, 2𝑜}} | |
3 | df-slt 31797 | . 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 31811 | . 2 ⊢ ¬ ∅ ∈ {1𝑜, 2𝑜} | |
5 | 1, 2, 3, 4 | soseq 31751 | 1 ⊢ <s Or No |
Colors of variables: wff setvar class |
Syntax hints: ∅c0 3915 {cpr 4179 {ctp 4181 〈cop 4183 Or wor 5034 1𝑜c1o 7553 2𝑜c2o 7554 No csur 31793 <s cslt 31794 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-1o 7560 df-2o 7561 df-no 31796 df-slt 31797 |
This theorem is referenced by: nosepne 31831 nosepdm 31834 nodenselem4 31837 nodenselem5 31838 nodenselem7 31840 nolt02o 31845 noresle 31846 nomaxmo 31847 noprefixmo 31848 nosupbnd1lem1 31854 nosupbnd1lem2 31855 nosupbnd1lem4 31857 nosupbnd1lem6 31859 nosupbnd1 31860 nosupbnd2lem1 31861 nosupbnd2 31862 noetalem3 31865 sltirr 31871 slttr 31872 sltasym 31873 sltlin 31874 slttrieq2 31875 slttrine 31876 sleloe 31879 sltletr 31881 slelttr 31882 |
Copyright terms: Public domain | W3C validator |