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Theorem nosep2o 27742
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 1136 . . . . . . . . . . . 12 ((𝐴 No 𝐵 No 𝐴𝐵) → 𝐵 No )
2 simp1 1135 . . . . . . . . . . . 12 ((𝐴 No 𝐵 No 𝐴𝐵) → 𝐴 No )
3 simp3 1137 . . . . . . . . . . . . 13 ((𝐴 No 𝐵 No 𝐴𝐵) → 𝐴𝐵)
43necomd 2994 . . . . . . . . . . . 12 ((𝐴 No 𝐵 No 𝐴𝐵) → 𝐵𝐴)
5 nosepne 27740 . . . . . . . . . . . 12 ((𝐵 No 𝐴 No 𝐵𝐴) → (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) ≠ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}))
61, 2, 4, 5syl3anc 1370 . . . . . . . . . . 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 3008 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) → (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) ≠ 2o)
109neneqd 2943 . . . . . . . 8 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) → ¬ (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o)
11 simpl2 1191 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) → 𝐵 No )
12 nofv 27717 . . . . . . . . 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 1485 . . . . . . . 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 1330 . . . . 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 1331 . . . . 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 6920 . . . 4 (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) ∈ V
25 fvex 6920 . . . 4 (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) ∈ V
2624, 25brtp 5533 . . 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 1190 . . 3 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) → 𝐴 No )
29 sltval2 27716 . . 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 1537  wcel 2106  wne 2938  {crab 3433  c0 4339  {ctp 4635  cop 4637   cint 4951   class class class wbr 5148  Oncon0 6386  cfv 6563  1oc1o 8498  2oc2o 8499   No csur 27699   <s cslt 27700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703
This theorem is referenced by:  noetainflem4  27800
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