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Theorem nosep2o 27628
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 1135 . . . . . . . . . . . 12 ((𝐴 No 𝐵 No 𝐴𝐵) → 𝐵 No )
2 simp1 1134 . . . . . . . . . . . 12 ((𝐴 No 𝐵 No 𝐴𝐵) → 𝐴 No )
3 simp3 1136 . . . . . . . . . . . . 13 ((𝐴 No 𝐵 No 𝐴𝐵) → 𝐴𝐵)
43necomd 2993 . . . . . . . . . . . 12 ((𝐴 No 𝐵 No 𝐴𝐵) → 𝐵𝐴)
5 nosepne 27626 . . . . . . . . . . . 12 ((𝐵 No 𝐴 No 𝐵𝐴) → (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) ≠ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}))
61, 2, 4, 5syl3anc 1369 . . . . . . . . . . 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 3007 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) → (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) ≠ 2o)
109neneqd 2942 . . . . . . . 8 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) → ¬ (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o)
11 simpl2 1190 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) → 𝐵 No )
12 nofv 27603 . . . . . . . . 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 1482 . . . . . . . 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 870 . . . . . 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 1007 . . . . 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 217 . . . 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 1329 . . . . 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 1330 . . . . 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 856 . . . 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 6910 . . . 4 (𝐵 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) ∈ V
25 fvex 6910 . . . 4 (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) ∈ V
2624, 25brtp 5525 . . 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 233 . 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 1189 . . 3 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐵𝑥) ≠ (𝐴𝑥)}) = 2o) → 𝐴 No )
29 sltval2 27602 . . 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 583 . 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 205  wa 395  wo 846  w3o 1084  w3a 1085   = wceq 1534  wcel 2099  wne 2937  {crab 3429  c0 4323  {ctp 4633  cop 4635   cint 4949   class class class wbr 5148  Oncon0 6369  cfv 6548  1oc1o 8480  2oc2o 8481   No csur 27586   <s cslt 27587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6372  df-on 6373  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-1o 8487  df-2o 8488  df-no 27589  df-slt 27590
This theorem is referenced by:  noetainflem4  27686
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