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

Theorem noextenddif 27178
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 27160 . . 3 (𝐴 No → dom 𝐴 ∈ On)
2 noextend.1 . . . . . 6 𝑋 ∈ {1o, 2o}
32nosgnn0i 27169 . . . . 5 ∅ ≠ 𝑋
43a1i 11 . . . 4 (𝐴 No → ∅ ≠ 𝑋)
5 nodmord 27163 . . . . . 6 (𝐴 No → Ord dom 𝐴)
6 ordirr 6382 . . . . . 6 (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
75, 6syl 17 . . . . 5 (𝐴 No → ¬ dom 𝐴 ∈ dom 𝐴)
8 ndmfv 6926 . . . . 5 (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
97, 8syl 17 . . . 4 (𝐴 No → (𝐴‘dom 𝐴) = ∅)
10 nofun 27159 . . . . . . 7 (𝐴 No → Fun 𝐴)
11 funfn 6578 . . . . . . 7 (Fun 𝐴𝐴 Fn dom 𝐴)
1210, 11sylib 217 . . . . . 6 (𝐴 No 𝐴 Fn dom 𝐴)
13 fnsng 6600 . . . . . . 7 ((dom 𝐴 ∈ On ∧ 𝑋 ∈ {1o, 2o}) → {⟨dom 𝐴, 𝑋⟩} Fn {dom 𝐴})
141, 2, 13sylancl 586 . . . . . 6 (𝐴 No → {⟨dom 𝐴, 𝑋⟩} Fn {dom 𝐴})
15 disjsn 4715 . . . . . . 7 ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
167, 15sylibr 233 . . . . . 6 (𝐴 No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
17 snidg 4662 . . . . . . 7 (dom 𝐴 ∈ On → dom 𝐴 ∈ {dom 𝐴})
181, 17syl 17 . . . . . 6 (𝐴 No → dom 𝐴 ∈ {dom 𝐴})
19 fvun2 6983 . . . . . 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 1374 . . . . 5 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴) = ({⟨dom 𝐴, 𝑋⟩}‘dom 𝐴))
21 fvsng 7180 . . . . . 6 ((dom 𝐴 ∈ On ∧ 𝑋 ∈ {1o, 2o}) → ({⟨dom 𝐴, 𝑋⟩}‘dom 𝐴) = 𝑋)
221, 2, 21sylancl 586 . . . . 5 (𝐴 No → ({⟨dom 𝐴, 𝑋⟩}‘dom 𝐴) = 𝑋)
2320, 22eqtrd 2772 . . . 4 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴) = 𝑋)
244, 9, 233netr4d 3018 . . 3 (𝐴 No → (𝐴‘dom 𝐴) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴))
25 fveq2 6891 . . . . 5 (𝑥 = dom 𝐴 → (𝐴𝑥) = (𝐴‘dom 𝐴))
26 fveq2 6891 . . . . 5 (𝑥 = dom 𝐴 → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴))
2725, 26neeq12d 3002 . . . 4 (𝑥 = dom 𝐴 → ((𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) ↔ (𝐴‘dom 𝐴) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴)))
2827onintss 6415 . . 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 6374 . . . . . . . 8 (𝑦 ∈ On → Ord 𝑦)
31 ordtri2 6399 . . . . . . . . . 10 ((Ord 𝑦 ∧ Ord dom 𝐴) → (𝑦 ∈ dom 𝐴 ↔ ¬ (𝑦 = dom 𝐴 ∨ dom 𝐴𝑦)))
32 eqcom 2739 . . . . . . . . . . . . 13 (𝑦 = dom 𝐴 ↔ dom 𝐴 = 𝑦)
3332orbi1i 912 . . . . . . . . . . . 12 ((𝑦 = dom 𝐴 ∨ dom 𝐴𝑦) ↔ (dom 𝐴 = 𝑦 ∨ dom 𝐴𝑦))
34 orcom 868 . . . . . . . . . . . 12 ((dom 𝐴 = 𝑦 ∨ dom 𝐴𝑦) ↔ (dom 𝐴𝑦 ∨ dom 𝐴 = 𝑦))
3533, 34bitri 274 . . . . . . . . . . 11 ((𝑦 = dom 𝐴 ∨ dom 𝐴𝑦) ↔ (dom 𝐴𝑦 ∨ dom 𝐴 = 𝑦))
3635notbii 319 . . . . . . . . . 10 (¬ (𝑦 = dom 𝐴 ∨ dom 𝐴𝑦) ↔ ¬ (dom 𝐴𝑦 ∨ dom 𝐴 = 𝑦))
3731, 36bitrdi 286 . . . . . . . . 9 ((Ord 𝑦 ∧ Ord dom 𝐴) → (𝑦 ∈ dom 𝐴 ↔ ¬ (dom 𝐴𝑦 ∨ dom 𝐴 = 𝑦)))
38 ordsseleq 6393 . . . . . . . . . . 11 ((Ord dom 𝐴 ∧ Ord 𝑦) → (dom 𝐴𝑦 ↔ (dom 𝐴𝑦 ∨ dom 𝐴 = 𝑦)))
3938notbid 317 . . . . . . . . . 10 ((Ord dom 𝐴 ∧ Ord 𝑦) → (¬ dom 𝐴𝑦 ↔ ¬ (dom 𝐴𝑦 ∨ dom 𝐴 = 𝑦)))
4039ancoms 459 . . . . . . . . 9 ((Ord 𝑦 ∧ Ord dom 𝐴) → (¬ dom 𝐴𝑦 ↔ ¬ (dom 𝐴𝑦 ∨ dom 𝐴 = 𝑦)))
4137, 40bitr4d 281 . . . . . . . 8 ((Ord 𝑦 ∧ Ord dom 𝐴) → (𝑦 ∈ dom 𝐴 ↔ ¬ dom 𝐴𝑦))
4230, 5, 41syl2anr 597 . . . . . . 7 ((𝐴 No 𝑦 ∈ On) → (𝑦 ∈ dom 𝐴 ↔ ¬ dom 𝐴𝑦))
43123ad2ant1 1133 . . . . . . . . . 10 ((𝐴 No 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → 𝐴 Fn dom 𝐴)
44143ad2ant1 1133 . . . . . . . . . 10 ((𝐴 No 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → {⟨dom 𝐴, 𝑋⟩} Fn {dom 𝐴})
45163ad2ant1 1133 . . . . . . . . . 10 ((𝐴 No 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
46 simp3 1138 . . . . . . . . . 10 ((𝐴 No 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → 𝑦 ∈ dom 𝐴)
47 fvun1 6982 . . . . . . . . . 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 1374 . . . . . . . . 9 ((𝐴 No 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) = (𝐴𝑦))
4948eqcomd 2738 . . . . . . . 8 ((𝐴 No 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → (𝐴𝑦) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦))
50493expia 1121 . . . . . . 7 ((𝐴 No 𝑦 ∈ On) → (𝑦 ∈ dom 𝐴 → (𝐴𝑦) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦)))
5142, 50sylbird 259 . . . . . 6 ((𝐴 No 𝑦 ∈ On) → (¬ dom 𝐴𝑦 → (𝐴𝑦) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦)))
5251necon1ad 2957 . . . . 5 ((𝐴 No 𝑦 ∈ On) → ((𝐴𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) → dom 𝐴𝑦))
5352ralrimiva 3146 . . . 4 (𝐴 No → ∀𝑦 ∈ On ((𝐴𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) → dom 𝐴𝑦))
54 fveq2 6891 . . . . . 6 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
55 fveq2 6891 . . . . . 6 (𝑥 = 𝑦 → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦))
5654, 55neeq12d 3002 . . . . 5 (𝑥 = 𝑦 → ((𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) ↔ (𝐴𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦)))
5756ralrab 3689 . . . 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 4968 . . 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 3999 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 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wne 2940  wral 3061  {crab 3432  cun 3946  cin 3947  wss 3948  c0 4322  {csn 4628  {cpr 4630  cop 4634   cint 4950  dom cdm 5676  Ord word 6363  Oncon0 6364  Fun wfun 6537   Fn wfn 6538  cfv 6543  1oc1o 8461  2oc2o 8462   No csur 27150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-1o 8468  df-2o 8469  df-no 27153
This theorem is referenced by:  noextendlt  27179  noextendgt  27180
  Copyright terms: Public domain W3C validator