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Theorem nosep1o 33621
Description: If the value of a surreal at a separator is 1o then the surreal is lesser. (Contributed by Scott Fenton, 7-Dec-2021.)
Assertion
Ref Expression
nosep1o (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o) → 𝐴 <s 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nosep1o
StepHypRef Expression
1 simpr 488 . . 3 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o) → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o)
2 nosepne 33620 . . . . . . . . . . . 12 ((𝐴 No 𝐵 No 𝐴𝐵) → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}))
32adantr 484 . . . . . . . . . . 11 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o) → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}))
41, 3eqnetrrd 3009 . . . . . . . . . 10 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o) → 1o ≠ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}))
54necomd 2996 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o) → (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ 1o)
65neneqd 2945 . . . . . . . 8 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o) → ¬ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o)
7 simpl2 1194 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o) → 𝐵 No )
8 nofv 33597 . . . . . . . . 9 (𝐵 No → ((𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅ ∨ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∨ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o))
97, 8syl 17 . . . . . . . 8 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o) → ((𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅ ∨ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∨ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o))
10 3orel2 33369 . . . . . . . 8 (¬ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o → (((𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅ ∨ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∨ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o) → ((𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅ ∨ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)))
116, 9, 10sylc 65 . . . . . . 7 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o) → ((𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅ ∨ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o))
12 eqid 2737 . . . . . . 7 1o = 1o
1311, 12jctil 523 . . . . . 6 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o) → (1o = 1o ∧ ((𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅ ∨ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)))
14 andi 1008 . . . . . 6 ((1o = 1o ∧ ((𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅ ∨ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)) ↔ ((1o = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅) ∨ (1o = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)))
1513, 14sylib 221 . . . . 5 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o) → ((1o = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅) ∨ (1o = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)))
16 3mix1 1332 . . . . . 6 ((1o = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅) → ((1o = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅) ∨ (1o = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o) ∨ (1o = ∅ ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)))
17 3mix2 1333 . . . . . 6 ((1o = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o) → ((1o = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅) ∨ (1o = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o) ∨ (1o = ∅ ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)))
1816, 17jaoi 857 . . . . 5 (((1o = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅) ∨ (1o = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)) → ((1o = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅) ∨ (1o = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o) ∨ (1o = ∅ ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)))
1915, 18syl 17 . . . 4 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o) → ((1o = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅) ∨ (1o = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o) ∨ (1o = ∅ ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)))
20 1oex 8215 . . . . 5 1o ∈ V
21 fvex 6730 . . . . 5 (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ∈ V
2220, 21brtp 33435 . . . 4 (1o{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ↔ ((1o = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅) ∨ (1o = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o) ∨ (1o = ∅ ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)))
2319, 22sylibr 237 . . 3 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o) → 1o{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}))
241, 23eqbrtrd 5075 . 2 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o) → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}))
25 simpl1 1193 . . 3 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o) → 𝐴 No )
26 sltval2 33596 . . 3 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)})))
2725, 7, 26syl2anc 587 . 2 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o) → (𝐴 <s 𝐵 ↔ (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)})))
2824, 27mpbird 260 1 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o) → 𝐴 <s 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 847  w3o 1088  w3a 1089   = wceq 1543  wcel 2110  wne 2940  {crab 3065  c0 4237  {ctp 4545  cop 4547   cint 4859   class class class wbr 5053  Oncon0 6213  cfv 6380  1oc1o 8195  2oc2o 8196   No csur 33580   <s cslt 33581
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 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
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 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-1o 8202  df-2o 8203  df-no 33583  df-slt 33584
This theorem is referenced by:  noetasuplem4  33676
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