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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. Alling's axiom (O). (Contributed by Scott Fenton, 9-Jun-2011.) |
Ref | Expression |
---|---|
sltso | ⊢ <s Or No |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sltsolem1 31067 | . 2 ⊢ {〈1𝑜, ∅〉, 〈1𝑜, 2𝑜〉, 〈∅, 2𝑜〉} Or ({1𝑜, 2𝑜} ∪ {∅}) | |
2 | df-no 31040 | . 2 ⊢ No = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶{1𝑜, 2𝑜}} | |
3 | df-slt 31041 | . 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 31055 | . 2 ⊢ ¬ ∅ ∈ {1𝑜, 2𝑜} | |
5 | 1, 2, 3, 4 | soseq 30995 | 1 ⊢ <s Or No |
Colors of variables: wff setvar class |
Syntax hints: ∅c0 3874 {cpr 4127 {ctp 4129 〈cop 4131 Or wor 4958 1𝑜c1o 7440 2𝑜c2o 7441 No csur 31037 <s cslt 31038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-1o 7447 df-2o 7448 df-no 31040 df-slt 31041 |
This theorem is referenced by: sltirr 31069 slttr 31070 slttri 31072 slttrieq2 31073 |
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