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Theorem noextenddif 33300
 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 33282 . . 3 (𝐴 No → dom 𝐴 ∈ On)
2 noextend.1 . . . . . 6 𝑋 ∈ {1o, 2o}
32nosgnn0i 33291 . . . . 5 ∅ ≠ 𝑋
43a1i 11 . . . 4 (𝐴 No → ∅ ≠ 𝑋)
5 nodmord 33285 . . . . . 6 (𝐴 No → Ord dom 𝐴)
6 ordirr 6177 . . . . . 6 (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
75, 6syl 17 . . . . 5 (𝐴 No → ¬ dom 𝐴 ∈ dom 𝐴)
8 ndmfv 6675 . . . . 5 (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
97, 8syl 17 . . . 4 (𝐴 No → (𝐴‘dom 𝐴) = ∅)
10 nofun 33281 . . . . . . 7 (𝐴 No → Fun 𝐴)
11 funfn 6354 . . . . . . 7 (Fun 𝐴𝐴 Fn dom 𝐴)
1210, 11sylib 221 . . . . . 6 (𝐴 No 𝐴 Fn dom 𝐴)
13 fnsng 6376 . . . . . . 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 4607 . . . . . . 7 ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
167, 15sylibr 237 . . . . . 6 (𝐴 No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
17 snidg 4559 . . . . . . 7 (dom 𝐴 ∈ On → dom 𝐴 ∈ {dom 𝐴})
181, 17syl 17 . . . . . 6 (𝐴 No → dom 𝐴 ∈ {dom 𝐴})
19 fvun2 6730 . . . . . 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 1371 . . . . 5 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴) = ({⟨dom 𝐴, 𝑋⟩}‘dom 𝐴))
21 fvsng 6919 . . . . . 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 2833 . . . 4 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴) = 𝑋)
244, 9, 233netr4d 3064 . . 3 (𝐴 No → (𝐴‘dom 𝐴) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴))
25 fveq2 6645 . . . . 5 (𝑥 = dom 𝐴 → (𝐴𝑥) = (𝐴‘dom 𝐴))
26 fveq2 6645 . . . . 5 (𝑥 = dom 𝐴 → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴))
2725, 26neeq12d 3048 . . . 4 (𝑥 = dom 𝐴 → ((𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) ↔ (𝐴‘dom 𝐴) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴)))
2827onintss 6209 . . 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 6169 . . . . . . . 8 (𝑦 ∈ On → Ord 𝑦)
31 ordtri2 6194 . . . . . . . . . 10 ((Ord 𝑦 ∧ Ord dom 𝐴) → (𝑦 ∈ dom 𝐴 ↔ ¬ (𝑦 = dom 𝐴 ∨ dom 𝐴𝑦)))
32 eqcom 2805 . . . . . . . . . . . . 13 (𝑦 = dom 𝐴 ↔ dom 𝐴 = 𝑦)
3332orbi1i 911 . . . . . . . . . . . 12 ((𝑦 = dom 𝐴 ∨ dom 𝐴𝑦) ↔ (dom 𝐴 = 𝑦 ∨ dom 𝐴𝑦))
34 orcom 867 . . . . . . . . . . . 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, 36syl6bb 290 . . . . . . . . 9 ((Ord 𝑦 ∧ Ord dom 𝐴) → (𝑦 ∈ dom 𝐴 ↔ ¬ (dom 𝐴𝑦 ∨ dom 𝐴 = 𝑦)))
38 ordsseleq 6188 . . . . . . . . . . 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 599 . . . . . . 7 ((𝐴 No 𝑦 ∈ On) → (𝑦 ∈ dom 𝐴 ↔ ¬ dom 𝐴𝑦))
43123ad2ant1 1130 . . . . . . . . . 10 ((𝐴 No 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → 𝐴 Fn dom 𝐴)
44143ad2ant1 1130 . . . . . . . . . 10 ((𝐴 No 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → {⟨dom 𝐴, 𝑋⟩} Fn {dom 𝐴})
45163ad2ant1 1130 . . . . . . . . . 10 ((𝐴 No 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
46 simp3 1135 . . . . . . . . . 10 ((𝐴 No 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → 𝑦 ∈ dom 𝐴)
47 fvun1 6729 . . . . . . . . . 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 1371 . . . . . . . . 9 ((𝐴 No 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) = (𝐴𝑦))
4948eqcomd 2804 . . . . . . . 8 ((𝐴 No 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → (𝐴𝑦) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦))
50493expia 1118 . . . . . . 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 3004 . . . . 5 ((𝐴 No 𝑦 ∈ On) → ((𝐴𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) → dom 𝐴𝑦))
5352ralrimiva 3149 . . . 4 (𝐴 No → ∀𝑦 ∈ On ((𝐴𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) → dom 𝐴𝑦))
54 fveq2 6645 . . . . . 6 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
55 fveq2 6645 . . . . . 6 (𝑥 = 𝑦 → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦))
5654, 55neeq12d 3048 . . . . 5 (𝑥 = 𝑦 → ((𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) ↔ (𝐴𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦)))
5756ralrab 3633 . . . 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 4854 . . 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 3932 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 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  {crab 3110   ∪ cun 3879   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  {csn 4525  {cpr 4527  ⟨cop 4531  ∩ cint 4838  dom cdm 5519  Ord word 6158  Oncon0 6159  Fun wfun 6318   Fn wfn 6319  ‘cfv 6324  1oc1o 8080  2oc2o 8081   No csur 33272 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-1o 8087  df-2o 8088  df-no 33275 This theorem is referenced by:  noextendlt  33301  noextendgt  33302
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