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Theorem sltso 31068
Description: Surreal less than totally orders the surreals. Alling's axiom (O). (Contributed by Scott Fenton, 9-Jun-2011.)
Assertion
Ref Expression
sltso <s Or No

Proof of Theorem sltso
Dummy variables 𝑓 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef 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 (∀𝑦𝑥 (𝑓𝑦) = (𝑔𝑦) ∧ (𝑓𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑔𝑥)))}
4 nosgnn0 31055 . 2 ¬ ∅ ∈ {1𝑜, 2𝑜}
51, 2, 3, 4soseq 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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