MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  noextenddif Structured version   Visualization version   GIF version

Theorem noextenddif 27171
Description: Calculate the place where a surreal and its extension differ. (Contributed by Scott Fenton, 22-Nov-2021.)
Hypothesis
Ref Expression
noextend.1 𝑋 ∈ {1o, 2o}
Assertion
Ref Expression
noextenddif (𝐴 No {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)} = dom 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑋

Proof of Theorem noextenddif
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nodmon 27153 . . 3 (𝐴 No → dom 𝐴 ∈ On)
2 noextend.1 . . . . . 6 𝑋 ∈ {1o, 2o}
32nosgnn0i 27162 . . . . 5 ∅ ≠ 𝑋
43a1i 11 . . . 4 (𝐴 No → ∅ ≠ 𝑋)
5 nodmord 27156 . . . . . 6 (𝐴 No → Ord dom 𝐴)
6 ordirr 6383 . . . . . 6 (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
75, 6syl 17 . . . . 5 (𝐴 No → ¬ dom 𝐴 ∈ dom 𝐴)
8 ndmfv 6927 . . . . 5 (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
97, 8syl 17 . . . 4 (𝐴 No → (𝐴‘dom 𝐴) = ∅)
10 nofun 27152 . . . . . . 7 (𝐴 No → Fun 𝐴)
11 funfn 6579 . . . . . . 7 (Fun 𝐴𝐴 Fn dom 𝐴)
1210, 11sylib 217 . . . . . 6 (𝐴 No 𝐴 Fn dom 𝐴)
13 fnsng 6601 . . . . . . 7 ((dom 𝐴 ∈ On ∧ 𝑋 ∈ {1o, 2o}) → {⟨dom 𝐴, 𝑋⟩} Fn {dom 𝐴})
141, 2, 13sylancl 587 . . . . . 6 (𝐴 No → {⟨dom 𝐴, 𝑋⟩} Fn {dom 𝐴})
15 disjsn 4716 . . . . . . 7 ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
167, 15sylibr 233 . . . . . 6 (𝐴 No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
17 snidg 4663 . . . . . . 7 (dom 𝐴 ∈ On → dom 𝐴 ∈ {dom 𝐴})
181, 17syl 17 . . . . . 6 (𝐴 No → dom 𝐴 ∈ {dom 𝐴})
19 fvun2 6984 . . . . . 6 ((𝐴 Fn dom 𝐴 ∧ {⟨dom 𝐴, 𝑋⟩} Fn {dom 𝐴} ∧ ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ∧ dom 𝐴 ∈ {dom 𝐴})) → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴) = ({⟨dom 𝐴, 𝑋⟩}‘dom 𝐴))
2012, 14, 16, 18, 19syl112anc 1375 . . . . 5 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴) = ({⟨dom 𝐴, 𝑋⟩}‘dom 𝐴))
21 fvsng 7178 . . . . . 6 ((dom 𝐴 ∈ On ∧ 𝑋 ∈ {1o, 2o}) → ({⟨dom 𝐴, 𝑋⟩}‘dom 𝐴) = 𝑋)
221, 2, 21sylancl 587 . . . . 5 (𝐴 No → ({⟨dom 𝐴, 𝑋⟩}‘dom 𝐴) = 𝑋)
2320, 22eqtrd 2773 . . . 4 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴) = 𝑋)
244, 9, 233netr4d 3019 . . 3 (𝐴 No → (𝐴‘dom 𝐴) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴))
25 fveq2 6892 . . . . 5 (𝑥 = dom 𝐴 → (𝐴𝑥) = (𝐴‘dom 𝐴))
26 fveq2 6892 . . . . 5 (𝑥 = dom 𝐴 → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴))
2725, 26neeq12d 3003 . . . 4 (𝑥 = dom 𝐴 → ((𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) ↔ (𝐴‘dom 𝐴) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴)))
2827onintss 6416 . . 3 (dom 𝐴 ∈ On → ((𝐴‘dom 𝐴) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)} ⊆ dom 𝐴))
291, 24, 28sylc 65 . 2 (𝐴 No {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)} ⊆ dom 𝐴)
30 eloni 6375 . . . . . . . 8 (𝑦 ∈ On → Ord 𝑦)
31 ordtri2 6400 . . . . . . . . . 10 ((Ord 𝑦 ∧ Ord dom 𝐴) → (𝑦 ∈ dom 𝐴 ↔ ¬ (𝑦 = dom 𝐴 ∨ dom 𝐴𝑦)))
32 eqcom 2740 . . . . . . . . . . . . 13 (𝑦 = dom 𝐴 ↔ dom 𝐴 = 𝑦)
3332orbi1i 913 . . . . . . . . . . . 12 ((𝑦 = dom 𝐴 ∨ dom 𝐴𝑦) ↔ (dom 𝐴 = 𝑦 ∨ dom 𝐴𝑦))
34 orcom 869 . . . . . . . . . . . 12 ((dom 𝐴 = 𝑦 ∨ dom 𝐴𝑦) ↔ (dom 𝐴𝑦 ∨ dom 𝐴 = 𝑦))
3533, 34bitri 275 . . . . . . . . . . 11 ((𝑦 = dom 𝐴 ∨ dom 𝐴𝑦) ↔ (dom 𝐴𝑦 ∨ dom 𝐴 = 𝑦))
3635notbii 320 . . . . . . . . . 10 (¬ (𝑦 = dom 𝐴 ∨ dom 𝐴𝑦) ↔ ¬ (dom 𝐴𝑦 ∨ dom 𝐴 = 𝑦))
3731, 36bitrdi 287 . . . . . . . . 9 ((Ord 𝑦 ∧ Ord dom 𝐴) → (𝑦 ∈ dom 𝐴 ↔ ¬ (dom 𝐴𝑦 ∨ dom 𝐴 = 𝑦)))
38 ordsseleq 6394 . . . . . . . . . . 11 ((Ord dom 𝐴 ∧ Ord 𝑦) → (dom 𝐴𝑦 ↔ (dom 𝐴𝑦 ∨ dom 𝐴 = 𝑦)))
3938notbid 318 . . . . . . . . . 10 ((Ord dom 𝐴 ∧ Ord 𝑦) → (¬ dom 𝐴𝑦 ↔ ¬ (dom 𝐴𝑦 ∨ dom 𝐴 = 𝑦)))
4039ancoms 460 . . . . . . . . 9 ((Ord 𝑦 ∧ Ord dom 𝐴) → (¬ dom 𝐴𝑦 ↔ ¬ (dom 𝐴𝑦 ∨ dom 𝐴 = 𝑦)))
4137, 40bitr4d 282 . . . . . . . 8 ((Ord 𝑦 ∧ Ord dom 𝐴) → (𝑦 ∈ dom 𝐴 ↔ ¬ dom 𝐴𝑦))
4230, 5, 41syl2anr 598 . . . . . . 7 ((𝐴 No 𝑦 ∈ On) → (𝑦 ∈ dom 𝐴 ↔ ¬ dom 𝐴𝑦))
43123ad2ant1 1134 . . . . . . . . . 10 ((𝐴 No 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → 𝐴 Fn dom 𝐴)
44143ad2ant1 1134 . . . . . . . . . 10 ((𝐴 No 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → {⟨dom 𝐴, 𝑋⟩} Fn {dom 𝐴})
45163ad2ant1 1134 . . . . . . . . . 10 ((𝐴 No 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
46 simp3 1139 . . . . . . . . . 10 ((𝐴 No 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → 𝑦 ∈ dom 𝐴)
47 fvun1 6983 . . . . . . . . . 10 ((𝐴 Fn dom 𝐴 ∧ {⟨dom 𝐴, 𝑋⟩} Fn {dom 𝐴} ∧ ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ∧ 𝑦 ∈ dom 𝐴)) → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) = (𝐴𝑦))
4843, 44, 45, 46, 47syl112anc 1375 . . . . . . . . 9 ((𝐴 No 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) = (𝐴𝑦))
4948eqcomd 2739 . . . . . . . 8 ((𝐴 No 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → (𝐴𝑦) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦))
50493expia 1122 . . . . . . 7 ((𝐴 No 𝑦 ∈ On) → (𝑦 ∈ dom 𝐴 → (𝐴𝑦) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦)))
5142, 50sylbird 260 . . . . . 6 ((𝐴 No 𝑦 ∈ On) → (¬ dom 𝐴𝑦 → (𝐴𝑦) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦)))
5251necon1ad 2958 . . . . 5 ((𝐴 No 𝑦 ∈ On) → ((𝐴𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) → dom 𝐴𝑦))
5352ralrimiva 3147 . . . 4 (𝐴 No → ∀𝑦 ∈ On ((𝐴𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) → dom 𝐴𝑦))
54 fveq2 6892 . . . . . 6 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
55 fveq2 6892 . . . . . 6 (𝑥 = 𝑦 → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦))
5654, 55neeq12d 3003 . . . . 5 (𝑥 = 𝑦 → ((𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) ↔ (𝐴𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦)))
5756ralrab 3690 . . . 4 (∀𝑦 ∈ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)}dom 𝐴𝑦 ↔ ∀𝑦 ∈ On ((𝐴𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) → dom 𝐴𝑦))
5853, 57sylibr 233 . . 3 (𝐴 No → ∀𝑦 ∈ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)}dom 𝐴𝑦)
59 ssint 4969 . . 3 (dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)} ↔ ∀𝑦 ∈ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)}dom 𝐴𝑦)
6058, 59sylibr 233 . 2 (𝐴 No → dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)})
6129, 60eqssd 4000 1 (𝐴 No {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)} = dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wne 2941  wral 3062  {crab 3433  cun 3947  cin 3948  wss 3949  c0 4323  {csn 4629  {cpr 4631  cop 4635   cint 4951  dom cdm 5677  Ord word 6364  Oncon0 6365  Fun wfun 6538   Fn wfn 6539  cfv 6544  1oc1o 8459  2oc2o 8460   No csur 27143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-1o 8466  df-2o 8467  df-no 27146
This theorem is referenced by:  noextendlt  27172  noextendgt  27173
  Copyright terms: Public domain W3C validator