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Theorem noextenddif 32142
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 32124 . . 3 (𝐴 No → dom 𝐴 ∈ On)
2 noextend.1 . . . . . 6 𝑋 ∈ {1𝑜, 2𝑜}
32nosgnn0i 32133 . . . . 5 ∅ ≠ 𝑋
43a1i 11 . . . 4 (𝐴 No → ∅ ≠ 𝑋)
5 nodmord 32127 . . . . . 6 (𝐴 No → Ord dom 𝐴)
6 ordirr 5954 . . . . . 6 (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
75, 6syl 17 . . . . 5 (𝐴 No → ¬ dom 𝐴 ∈ dom 𝐴)
8 ndmfv 6438 . . . . 5 (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
97, 8syl 17 . . . 4 (𝐴 No → (𝐴‘dom 𝐴) = ∅)
10 nofun 32123 . . . . . . 7 (𝐴 No → Fun 𝐴)
11 funfn 6131 . . . . . . 7 (Fun 𝐴𝐴 Fn dom 𝐴)
1210, 11sylib 209 . . . . . 6 (𝐴 No 𝐴 Fn dom 𝐴)
13 fnsng 6152 . . . . . . 7 ((dom 𝐴 ∈ On ∧ 𝑋 ∈ {1𝑜, 2𝑜}) → {⟨dom 𝐴, 𝑋⟩} Fn {dom 𝐴})
141, 2, 13sylancl 576 . . . . . 6 (𝐴 No → {⟨dom 𝐴, 𝑋⟩} Fn {dom 𝐴})
15 disjsn 4438 . . . . . . 7 ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
167, 15sylibr 225 . . . . . 6 (𝐴 No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
17 snidg 4400 . . . . . . 7 (dom 𝐴 ∈ On → dom 𝐴 ∈ {dom 𝐴})
181, 17syl 17 . . . . . 6 (𝐴 No → dom 𝐴 ∈ {dom 𝐴})
19 fvun2 6491 . . . . . 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 1486 . . . . 5 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴) = ({⟨dom 𝐴, 𝑋⟩}‘dom 𝐴))
21 fvsng 6672 . . . . . 6 ((dom 𝐴 ∈ On ∧ 𝑋 ∈ {1𝑜, 2𝑜}) → ({⟨dom 𝐴, 𝑋⟩}‘dom 𝐴) = 𝑋)
221, 2, 21sylancl 576 . . . . 5 (𝐴 No → ({⟨dom 𝐴, 𝑋⟩}‘dom 𝐴) = 𝑋)
2320, 22eqtrd 2840 . . . 4 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴) = 𝑋)
244, 9, 233netr4d 3055 . . 3 (𝐴 No → (𝐴‘dom 𝐴) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴))
25 fveq2 6408 . . . . 5 (𝑥 = dom 𝐴 → (𝐴𝑥) = (𝐴‘dom 𝐴))
26 fveq2 6408 . . . . 5 (𝑥 = dom 𝐴 → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴))
2725, 26neeq12d 3039 . . . 4 (𝑥 = dom 𝐴 → ((𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) ↔ (𝐴‘dom 𝐴) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴)))
2827onintss 5988 . . 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 5946 . . . . . . . 8 (𝑦 ∈ On → Ord 𝑦)
31 ordtri2 5971 . . . . . . . . . 10 ((Ord 𝑦 ∧ Ord dom 𝐴) → (𝑦 ∈ dom 𝐴 ↔ ¬ (𝑦 = dom 𝐴 ∨ dom 𝐴𝑦)))
32 eqcom 2813 . . . . . . . . . . . . 13 (𝑦 = dom 𝐴 ↔ dom 𝐴 = 𝑦)
3332orbi1i 928 . . . . . . . . . . . 12 ((𝑦 = dom 𝐴 ∨ dom 𝐴𝑦) ↔ (dom 𝐴 = 𝑦 ∨ dom 𝐴𝑦))
34 orcom 888 . . . . . . . . . . . 12 ((dom 𝐴 = 𝑦 ∨ dom 𝐴𝑦) ↔ (dom 𝐴𝑦 ∨ dom 𝐴 = 𝑦))
3533, 34bitri 266 . . . . . . . . . . 11 ((𝑦 = dom 𝐴 ∨ dom 𝐴𝑦) ↔ (dom 𝐴𝑦 ∨ dom 𝐴 = 𝑦))
3635notbii 311 . . . . . . . . . 10 (¬ (𝑦 = dom 𝐴 ∨ dom 𝐴𝑦) ↔ ¬ (dom 𝐴𝑦 ∨ dom 𝐴 = 𝑦))
3731, 36syl6bb 278 . . . . . . . . 9 ((Ord 𝑦 ∧ Ord dom 𝐴) → (𝑦 ∈ dom 𝐴 ↔ ¬ (dom 𝐴𝑦 ∨ dom 𝐴 = 𝑦)))
38 ordsseleq 5965 . . . . . . . . . . 11 ((Ord dom 𝐴 ∧ Ord 𝑦) → (dom 𝐴𝑦 ↔ (dom 𝐴𝑦 ∨ dom 𝐴 = 𝑦)))
3938notbid 309 . . . . . . . . . 10 ((Ord dom 𝐴 ∧ Ord 𝑦) → (¬ dom 𝐴𝑦 ↔ ¬ (dom 𝐴𝑦 ∨ dom 𝐴 = 𝑦)))
4039ancoms 448 . . . . . . . . 9 ((Ord 𝑦 ∧ Ord dom 𝐴) → (¬ dom 𝐴𝑦 ↔ ¬ (dom 𝐴𝑦 ∨ dom 𝐴 = 𝑦)))
4137, 40bitr4d 273 . . . . . . . 8 ((Ord 𝑦 ∧ Ord dom 𝐴) → (𝑦 ∈ dom 𝐴 ↔ ¬ dom 𝐴𝑦))
4230, 5, 41syl2anr 586 . . . . . . 7 ((𝐴 No 𝑦 ∈ On) → (𝑦 ∈ dom 𝐴 ↔ ¬ dom 𝐴𝑦))
43123ad2ant1 1156 . . . . . . . . . 10 ((𝐴 No 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → 𝐴 Fn dom 𝐴)
44143ad2ant1 1156 . . . . . . . . . 10 ((𝐴 No 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → {⟨dom 𝐴, 𝑋⟩} Fn {dom 𝐴})
45163ad2ant1 1156 . . . . . . . . . 10 ((𝐴 No 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
46 simp3 1161 . . . . . . . . . 10 ((𝐴 No 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → 𝑦 ∈ dom 𝐴)
47 fvun1 6490 . . . . . . . . . 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 1486 . . . . . . . . 9 ((𝐴 No 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) = (𝐴𝑦))
4948eqcomd 2812 . . . . . . . 8 ((𝐴 No 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → (𝐴𝑦) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦))
50493expia 1143 . . . . . . 7 ((𝐴 No 𝑦 ∈ On) → (𝑦 ∈ dom 𝐴 → (𝐴𝑦) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦)))
5142, 50sylbird 251 . . . . . 6 ((𝐴 No 𝑦 ∈ On) → (¬ dom 𝐴𝑦 → (𝐴𝑦) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦)))
5251necon1ad 2995 . . . . 5 ((𝐴 No 𝑦 ∈ On) → ((𝐴𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) → dom 𝐴𝑦))
5352ralrimiva 3154 . . . 4 (𝐴 No → ∀𝑦 ∈ On ((𝐴𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) → dom 𝐴𝑦))
54 fveq2 6408 . . . . . 6 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
55 fveq2 6408 . . . . . 6 (𝑥 = 𝑦 → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦))
5654, 55neeq12d 3039 . . . . 5 (𝑥 = 𝑦 → ((𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) ↔ (𝐴𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦)))
5756ralrab 3564 . . . 4 (∀𝑦 ∈ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)}dom 𝐴𝑦 ↔ ∀𝑦 ∈ On ((𝐴𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) → dom 𝐴𝑦))
5853, 57sylibr 225 . . 3 (𝐴 No → ∀𝑦 ∈ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)}dom 𝐴𝑦)
59 ssint 4685 . . 3 (dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)} ↔ ∀𝑦 ∈ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)}dom 𝐴𝑦)
6058, 59sylibr 225 . 2 (𝐴 No → dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)})
6129, 60eqssd 3815 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 197  wa 384  wo 865  w3a 1100   = wceq 1637  wcel 2156  wne 2978  wral 3096  {crab 3100  cun 3767  cin 3768  wss 3769  c0 4116  {csn 4370  {cpr 4372  cop 4376   cint 4669  dom cdm 5311  Ord word 5935  Oncon0 5936  Fun wfun 6095   Fn wfn 6096  cfv 6101  1𝑜c1o 7789  2𝑜c2o 7790   No csur 32114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-ord 5939  df-on 5940  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-1o 7796  df-2o 7797  df-no 32117
This theorem is referenced by:  noextendlt  32143  noextendgt  32144
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