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Theorem nosep2o 27728
Description: If the value of a surreal at a separator is 2o then the surreal is greater. (Contributed by Scott Fenton, 7-Dec-2021.)
Assertion
Ref Expression
nosep2o (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) → 𝐵 <s 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nosep2o
StepHypRef Expression
1 simp2 1137 . . . . . . . . . . . 12 ((𝐴 No 𝐵 No 𝐴𝐵) → 𝐵 No )
2 simp1 1136 . . . . . . . . . . . 12 ((𝐴 No 𝐵 No 𝐴𝐵) → 𝐴 No )
3 simp3 1138 . . . . . . . . . . . . 13 ((𝐴 No 𝐵 No 𝐴𝐵) → 𝐴𝐵)
43necomd 2995 . . . . . . . . . . . 12 ((𝐴 No 𝐵 No 𝐴𝐵) → 𝐵𝐴)
5 nosepne 27726 . . . . . . . . . . . 12 ((𝐵 No 𝐴 No 𝐵𝐴) → (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) ≠ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}))
61, 2, 4, 5syl3anc 1372 . . . . . . . . . . 11 ((𝐴 No 𝐵 No 𝐴𝐵) → (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) ≠ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}))
76adantr 480 . . . . . . . . . 10 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) → (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) ≠ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}))
8 simpr 484 . . . . . . . . . 10 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) → (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o)
97, 8neeqtrd 3009 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) → (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) ≠ 2o)
109neneqd 2944 . . . . . . . 8 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) → ¬ (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o)
11 simpl2 1192 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) → 𝐵 No )
12 nofv 27703 . . . . . . . . 9 (𝐵 No → ((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = ∅ ∨ (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 1o ∨ (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o))
1311, 12syl 17 . . . . . . . 8 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) → ((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = ∅ ∨ (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 1o ∨ (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o))
14 3orel3 1487 . . . . . . . 8 (¬ (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o → (((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = ∅ ∨ (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 1o ∨ (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) → ((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = ∅ ∨ (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 1o)))
1510, 13, 14sylc 65 . . . . . . 7 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) → ((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = ∅ ∨ (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 1o))
1615orcomd 871 . . . . . 6 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) → ((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 1o ∨ (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = ∅))
1716, 8jca 511 . . . . 5 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) → (((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 1o ∨ (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = ∅) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o))
18 andir 1010 . . . . 5 ((((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 1o ∨ (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = ∅) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) ↔ (((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 1o ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) ∨ ((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = ∅ ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o)))
1917, 18sylib 218 . . . 4 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) → (((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 1o ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) ∨ ((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = ∅ ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o)))
20 3mix2 1331 . . . . 5 (((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 1o ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) → (((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 1o ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = ∅) ∨ ((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 1o ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) ∨ ((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = ∅ ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o)))
21 3mix3 1332 . . . . 5 (((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = ∅ ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) → (((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 1o ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = ∅) ∨ ((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 1o ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) ∨ ((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = ∅ ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o)))
2220, 21jaoi 857 . . . 4 ((((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 1o ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) ∨ ((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = ∅ ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o)) → (((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 1o ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = ∅) ∨ ((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 1o ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) ∨ ((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = ∅ ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o)))
2319, 22syl 17 . . 3 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) → (((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 1o ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = ∅) ∨ ((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 1o ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) ∨ ((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = ∅ ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o)))
24 fvex 6918 . . . 4 (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) ∈ V
25 fvex 6918 . . . 4 (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) ∈ V
2624, 25brtp 5527 . . 3 ((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) ↔ (((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 1o ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = ∅) ∨ ((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 1o ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) ∨ ((𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = ∅ ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o)))
2723, 26sylibr 234 . 2 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) → (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}))
28 simpl1 1191 . . 3 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) → 𝐴 No )
29 sltval2 27702 . . 3 ((𝐵 No 𝐴 No ) → (𝐵 <s 𝐴 ↔ (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)})))
3011, 28, 29syl2anc 584 . 2 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) → (𝐵 <s 𝐴 ↔ (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)})))
3127, 30mpbird 257 1 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) → 𝐵 <s 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  w3o 1085  w3a 1086   = wceq 1539  wcel 2107  wne 2939  {crab 3435  c0 4332  {ctp 4629  cop 4631   cint 4945   class class class wbr 5142  Oncon0 6383  cfv 6560  1oc1o 8500  2oc2o 8501   No csur 27685   <s cslt 27686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-1o 8507  df-2o 8508  df-no 27688  df-slt 27689
This theorem is referenced by:  noetainflem4  27786
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