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Theorem noextenddif 33608
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 33590 . . 3 (𝐴 No → dom 𝐴 ∈ On)
2 noextend.1 . . . . . 6 𝑋 ∈ {1o, 2o}
32nosgnn0i 33599 . . . . 5 ∅ ≠ 𝑋
43a1i 11 . . . 4 (𝐴 No → ∅ ≠ 𝑋)
5 nodmord 33593 . . . . . 6 (𝐴 No → Ord dom 𝐴)
6 ordirr 6231 . . . . . 6 (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
75, 6syl 17 . . . . 5 (𝐴 No → ¬ dom 𝐴 ∈ dom 𝐴)
8 ndmfv 6747 . . . . 5 (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
97, 8syl 17 . . . 4 (𝐴 No → (𝐴‘dom 𝐴) = ∅)
10 nofun 33589 . . . . . . 7 (𝐴 No → Fun 𝐴)
11 funfn 6410 . . . . . . 7 (Fun 𝐴𝐴 Fn dom 𝐴)
1210, 11sylib 221 . . . . . 6 (𝐴 No 𝐴 Fn dom 𝐴)
13 fnsng 6432 . . . . . . 7 ((dom 𝐴 ∈ On ∧ 𝑋 ∈ {1o, 2o}) → {⟨dom 𝐴, 𝑋⟩} Fn {dom 𝐴})
141, 2, 13sylancl 589 . . . . . 6 (𝐴 No → {⟨dom 𝐴, 𝑋⟩} Fn {dom 𝐴})
15 disjsn 4627 . . . . . . 7 ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
167, 15sylibr 237 . . . . . 6 (𝐴 No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
17 snidg 4575 . . . . . . 7 (dom 𝐴 ∈ On → dom 𝐴 ∈ {dom 𝐴})
181, 17syl 17 . . . . . 6 (𝐴 No → dom 𝐴 ∈ {dom 𝐴})
19 fvun2 6803 . . . . . 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 1376 . . . . 5 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴) = ({⟨dom 𝐴, 𝑋⟩}‘dom 𝐴))
21 fvsng 6995 . . . . . 6 ((dom 𝐴 ∈ On ∧ 𝑋 ∈ {1o, 2o}) → ({⟨dom 𝐴, 𝑋⟩}‘dom 𝐴) = 𝑋)
221, 2, 21sylancl 589 . . . . 5 (𝐴 No → ({⟨dom 𝐴, 𝑋⟩}‘dom 𝐴) = 𝑋)
2320, 22eqtrd 2777 . . . 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 6717 . . . . 5 (𝑥 = dom 𝐴 → (𝐴𝑥) = (𝐴‘dom 𝐴))
26 fveq2 6717 . . . . 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 6263 . . 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 6223 . . . . . . . 8 (𝑦 ∈ On → Ord 𝑦)
31 ordtri2 6248 . . . . . . . . . 10 ((Ord 𝑦 ∧ Ord dom 𝐴) → (𝑦 ∈ dom 𝐴 ↔ ¬ (𝑦 = dom 𝐴 ∨ dom 𝐴𝑦)))
32 eqcom 2744 . . . . . . . . . . . . 13 (𝑦 = dom 𝐴 ↔ dom 𝐴 = 𝑦)
3332orbi1i 914 . . . . . . . . . . . 12 ((𝑦 = dom 𝐴 ∨ dom 𝐴𝑦) ↔ (dom 𝐴 = 𝑦 ∨ dom 𝐴𝑦))
34 orcom 870 . . . . . . . . . . . 12 ((dom 𝐴 = 𝑦 ∨ dom 𝐴𝑦) ↔ (dom 𝐴𝑦 ∨ dom 𝐴 = 𝑦))
3533, 34bitri 278 . . . . . . . . . . 11 ((𝑦 = dom 𝐴 ∨ dom 𝐴𝑦) ↔ (dom 𝐴𝑦 ∨ dom 𝐴 = 𝑦))
3635notbii 323 . . . . . . . . . 10 (¬ (𝑦 = dom 𝐴 ∨ dom 𝐴𝑦) ↔ ¬ (dom 𝐴𝑦 ∨ dom 𝐴 = 𝑦))
3731, 36bitrdi 290 . . . . . . . . 9 ((Ord 𝑦 ∧ Ord dom 𝐴) → (𝑦 ∈ dom 𝐴 ↔ ¬ (dom 𝐴𝑦 ∨ dom 𝐴 = 𝑦)))
38 ordsseleq 6242 . . . . . . . . . . 11 ((Ord dom 𝐴 ∧ Ord 𝑦) → (dom 𝐴𝑦 ↔ (dom 𝐴𝑦 ∨ dom 𝐴 = 𝑦)))
3938notbid 321 . . . . . . . . . 10 ((Ord dom 𝐴 ∧ Ord 𝑦) → (¬ dom 𝐴𝑦 ↔ ¬ (dom 𝐴𝑦 ∨ dom 𝐴 = 𝑦)))
4039ancoms 462 . . . . . . . . 9 ((Ord 𝑦 ∧ Ord dom 𝐴) → (¬ dom 𝐴𝑦 ↔ ¬ (dom 𝐴𝑦 ∨ dom 𝐴 = 𝑦)))
4137, 40bitr4d 285 . . . . . . . 8 ((Ord 𝑦 ∧ Ord dom 𝐴) → (𝑦 ∈ dom 𝐴 ↔ ¬ dom 𝐴𝑦))
4230, 5, 41syl2anr 600 . . . . . . 7 ((𝐴 No 𝑦 ∈ On) → (𝑦 ∈ dom 𝐴 ↔ ¬ dom 𝐴𝑦))
43123ad2ant1 1135 . . . . . . . . . 10 ((𝐴 No 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → 𝐴 Fn dom 𝐴)
44143ad2ant1 1135 . . . . . . . . . 10 ((𝐴 No 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → {⟨dom 𝐴, 𝑋⟩} Fn {dom 𝐴})
45163ad2ant1 1135 . . . . . . . . . 10 ((𝐴 No 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
46 simp3 1140 . . . . . . . . . 10 ((𝐴 No 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → 𝑦 ∈ dom 𝐴)
47 fvun1 6802 . . . . . . . . . 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 1376 . . . . . . . . 9 ((𝐴 No 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) = (𝐴𝑦))
4948eqcomd 2743 . . . . . . . 8 ((𝐴 No 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → (𝐴𝑦) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦))
50493expia 1123 . . . . . . 7 ((𝐴 No 𝑦 ∈ On) → (𝑦 ∈ dom 𝐴 → (𝐴𝑦) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦)))
5142, 50sylbird 263 . . . . . 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 3105 . . . 4 (𝐴 No → ∀𝑦 ∈ On ((𝐴𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) → dom 𝐴𝑦))
54 fveq2 6717 . . . . . 6 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
55 fveq2 6717 . . . . . 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 3607 . . . 4 (∀𝑦 ∈ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)}dom 𝐴𝑦 ↔ ∀𝑦 ∈ On ((𝐴𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) → dom 𝐴𝑦))
5853, 57sylibr 237 . . 3 (𝐴 No → ∀𝑦 ∈ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)}dom 𝐴𝑦)
59 ssint 4875 . . 3 (dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)} ↔ ∀𝑦 ∈ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)}dom 𝐴𝑦)
6058, 59sylibr 237 . 2 (𝐴 No → dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)})
6129, 60eqssd 3918 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 209  wa 399  wo 847  w3a 1089   = wceq 1543  wcel 2110  wne 2940  wral 3061  {crab 3065  cun 3864  cin 3865  wss 3866  c0 4237  {csn 4541  {cpr 4543  cop 4547   cint 4859  dom cdm 5551  Ord word 6212  Oncon0 6213  Fun wfun 6374   Fn wfn 6375  cfv 6380  1oc1o 8195  2oc2o 8196   No csur 33580
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-1o 8202  df-2o 8203  df-no 33583
This theorem is referenced by:  noextendlt  33609  noextendgt  33610
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