Theorem noextenddif 27632

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 ∣ (𝐴‘𝑥) ≠ ((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.

Step	Hyp	Ref	Expression
1		nodmon 27614	. . 3 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
2		noextend.1	. . . . . 6 ⊢ 𝑋 ∈ {1o, 2o}
3	2	nosgnn0i 27623	. . . . 5 ⊢ ∅ ≠ 𝑋
4	3	a1i 11	. . . 4 ⊢ (𝐴 ∈ No → ∅ ≠ 𝑋)
5		nodmord 27617	. . . . . 6 ⊢ (𝐴 ∈ No → Ord dom 𝐴)
6		ordirr 6370	. . . . . 6 ⊢ (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
7	5, 6	syl 17	. . . . 5 ⊢ (𝐴 ∈ No → ¬ dom 𝐴 ∈ dom 𝐴)
8		ndmfv 6911	. . . . 5 ⊢ (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
9	7, 8	syl 17	. . . 4 ⊢ (𝐴 ∈ No → (𝐴‘dom 𝐴) = ∅)
10		nofun 27613	. . . . . . 7 ⊢ (𝐴 ∈ No → Fun 𝐴)
11		funfn 6566	. . . . . . 7 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
12	10, 11	sylib 218	. . . . . 6 ⊢ (𝐴 ∈ No → 𝐴 Fn dom 𝐴)
13		fnsng 6588	. . . . . . 7 ⊢ ((dom 𝐴 ∈ On ∧ 𝑋 ∈ {1o, 2o}) → {⟨dom 𝐴, 𝑋⟩} Fn {dom 𝐴})
14	1, 2, 13	sylancl 586	. . . . . 6 ⊢ (𝐴 ∈ No → {⟨dom 𝐴, 𝑋⟩} Fn {dom 𝐴})
15		disjsn 4687	. . . . . . 7 ⊢ ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
16	7, 15	sylibr 234	. . . . . 6 ⊢ (𝐴 ∈ No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
17		snidg 4636	. . . . . . 7 ⊢ (dom 𝐴 ∈ On → dom 𝐴 ∈ {dom 𝐴})
18	1, 17	syl 17	. . . . . 6 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ {dom 𝐴})
19		fvun2 6971	. . . . . 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))
20	12, 14, 16, 18, 19	syl112anc 1376	. . . . 5 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴) = ({⟨dom 𝐴, 𝑋⟩}‘dom 𝐴))
21		fvsng 7172	. . . . . 6 ⊢ ((dom 𝐴 ∈ On ∧ 𝑋 ∈ {1o, 2o}) → ({⟨dom 𝐴, 𝑋⟩}‘dom 𝐴) = 𝑋)
22	1, 2, 21	sylancl 586	. . . . 5 ⊢ (𝐴 ∈ No → ({⟨dom 𝐴, 𝑋⟩}‘dom 𝐴) = 𝑋)
23	20, 22	eqtrd 2770	. . . 4 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴) = 𝑋)
24	4, 9, 23	3netr4d 3009	. . 3 ⊢ (𝐴 ∈ No → (𝐴‘dom 𝐴) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴))
25		fveq2 6876	. . . . 5 ⊢ (𝑥 = dom 𝐴 → (𝐴‘𝑥) = (𝐴‘dom 𝐴))
26		fveq2 6876	. . . . 5 ⊢ (𝑥 = dom 𝐴 → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴))
27	25, 26	neeq12d 2993	. . . 4 ⊢ (𝑥 = dom 𝐴 → ((𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) ↔ (𝐴‘dom 𝐴) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴)))
28	27	onintss 6404	. . 3 ⊢ (dom 𝐴 ∈ On → ((𝐴‘dom 𝐴) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)} ⊆ dom 𝐴))
29	1, 24, 28	sylc 65	. 2 ⊢ (𝐴 ∈ No → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)} ⊆ dom 𝐴)
30		eloni 6362	. . . . . . . 8 ⊢ (𝑦 ∈ On → Ord 𝑦)
31		ordtri2 6387	. . . . . . . . . 10 ⊢ ((Ord 𝑦 ∧ Ord dom 𝐴) → (𝑦 ∈ dom 𝐴 ↔ ¬ (𝑦 = dom 𝐴 ∨ dom 𝐴 ∈ 𝑦)))
32		eqcom 2742	. . . . . . . . . . . . 13 ⊢ (𝑦 = dom 𝐴 ↔ dom 𝐴 = 𝑦)
33	32	orbi1i 913	. . . . . . . . . . . 12 ⊢ ((𝑦 = dom 𝐴 ∨ dom 𝐴 ∈ 𝑦) ↔ (dom 𝐴 = 𝑦 ∨ dom 𝐴 ∈ 𝑦))
34		orcom 870	. . . . . . . . . . . 12 ⊢ ((dom 𝐴 = 𝑦 ∨ dom 𝐴 ∈ 𝑦) ↔ (dom 𝐴 ∈ 𝑦 ∨ dom 𝐴 = 𝑦))
35	33, 34	bitri 275	. . . . . . . . . . 11 ⊢ ((𝑦 = dom 𝐴 ∨ dom 𝐴 ∈ 𝑦) ↔ (dom 𝐴 ∈ 𝑦 ∨ dom 𝐴 = 𝑦))
36	35	notbii 320	. . . . . . . . . 10 ⊢ (¬ (𝑦 = dom 𝐴 ∨ dom 𝐴 ∈ 𝑦) ↔ ¬ (dom 𝐴 ∈ 𝑦 ∨ dom 𝐴 = 𝑦))
37	31, 36	bitrdi 287	. . . . . . . . 9 ⊢ ((Ord 𝑦 ∧ Ord dom 𝐴) → (𝑦 ∈ dom 𝐴 ↔ ¬ (dom 𝐴 ∈ 𝑦 ∨ dom 𝐴 = 𝑦)))
38		ordsseleq 6381	. . . . . . . . . . 11 ⊢ ((Ord dom 𝐴 ∧ Ord 𝑦) → (dom 𝐴 ⊆ 𝑦 ↔ (dom 𝐴 ∈ 𝑦 ∨ dom 𝐴 = 𝑦)))
39	38	notbid 318	. . . . . . . . . 10 ⊢ ((Ord dom 𝐴 ∧ Ord 𝑦) → (¬ dom 𝐴 ⊆ 𝑦 ↔ ¬ (dom 𝐴 ∈ 𝑦 ∨ dom 𝐴 = 𝑦)))
40	39	ancoms 458	. . . . . . . . 9 ⊢ ((Ord 𝑦 ∧ Ord dom 𝐴) → (¬ dom 𝐴 ⊆ 𝑦 ↔ ¬ (dom 𝐴 ∈ 𝑦 ∨ dom 𝐴 = 𝑦)))
41	37, 40	bitr4d 282	. . . . . . . 8 ⊢ ((Ord 𝑦 ∧ Ord dom 𝐴) → (𝑦 ∈ dom 𝐴 ↔ ¬ dom 𝐴 ⊆ 𝑦))
42	30, 5, 41	syl2anr 597	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On) → (𝑦 ∈ dom 𝐴 ↔ ¬ dom 𝐴 ⊆ 𝑦))
43	12	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → 𝐴 Fn dom 𝐴)
44	14	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → {⟨dom 𝐴, 𝑋⟩} Fn {dom 𝐴})
45	16	3ad2ant1 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 6970	. . . . . . . . . 10 ⊢ ((𝐴 Fn dom 𝐴 ∧ {⟨dom 𝐴, 𝑋⟩} Fn {dom 𝐴} ∧ ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ∧ 𝑦 ∈ dom 𝐴)) → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) = (𝐴‘𝑦))
48	43, 44, 45, 46, 47	syl112anc 1376	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) = (𝐴‘𝑦))
49	48	eqcomd 2741	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → (𝐴‘𝑦) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦))
50	49	3expia 1121	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On) → (𝑦 ∈ dom 𝐴 → (𝐴‘𝑦) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦)))
51	42, 50	sylbird 260	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On) → (¬ dom 𝐴 ⊆ 𝑦 → (𝐴‘𝑦) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦)))
52	51	necon1ad 2949	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On) → ((𝐴‘𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) → dom 𝐴 ⊆ 𝑦))
53	52	ralrimiva 3132	. . . 4 ⊢ (𝐴 ∈ No → ∀𝑦 ∈ On ((𝐴‘𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) → dom 𝐴 ⊆ 𝑦))
54		fveq2 6876	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴‘𝑥) = (𝐴‘𝑦))
55		fveq2 6876	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦))
56	54, 55	neeq12d 2993	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) ↔ (𝐴‘𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦)))
57	56	ralrab 3677	. . . 4 ⊢ (∀𝑦 ∈ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)}dom 𝐴 ⊆ 𝑦 ↔ ∀𝑦 ∈ On ((𝐴‘𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) → dom 𝐴 ⊆ 𝑦))
58	53, 57	sylibr 234	. . 3 ⊢ (𝐴 ∈ No → ∀𝑦 ∈ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)}dom 𝐴 ⊆ 𝑦)
59		ssint 4940	. . 3 ⊢ (dom 𝐴 ⊆ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)} ↔ ∀𝑦 ∈ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)}dom 𝐴 ⊆ 𝑦)
60	58, 59	sylibr 234	. 2 ⊢ (𝐴 ∈ No → dom 𝐴 ⊆ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)})
61	29, 60	eqssd 3976	1 ⊢ (𝐴 ∈ No → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)} = dom 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  {crab 3415   ∪ cun 3924   ∩ cin 3925   ⊆ wss 3926  ∅c0 4308  {csn 4601  {cpr 4603  ⟨cop 4607  ∩ cint 4922  dom cdm 5654  Ord word 6351  Oncon0 6352  Fun wfun 6525   Fn wfn 6526  ‘cfv 6531  1_oc1o 8473  2_oc2o 8474   No csur 27603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-br 5120  df-opab 5182  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ord 6355  df-on 6356  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-1o 8480  df-2o 8481  df-no 27606

This theorem is referenced by:  noextendlt  27633  noextendgt  27634