Theorem noextenddif 27161

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 27143	. . 3 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
2		noextend.1	. . . . . 6 ⊢ 𝑋 ∈ {1o, 2o}
3	2	nosgnn0i 27152	. . . . 5 ⊢ ∅ ≠ 𝑋
4	3	a1i 11	. . . 4 ⊢ (𝐴 ∈ No → ∅ ≠ 𝑋)
5		nodmord 27146	. . . . . 6 ⊢ (𝐴 ∈ No → Ord dom 𝐴)
6		ordirr 6380	. . . . . 6 ⊢ (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
7	5, 6	syl 17	. . . . 5 ⊢ (𝐴 ∈ No → ¬ dom 𝐴 ∈ dom 𝐴)
8		ndmfv 6924	. . . . 5 ⊢ (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
9	7, 8	syl 17	. . . 4 ⊢ (𝐴 ∈ No → (𝐴‘dom 𝐴) = ∅)
10		nofun 27142	. . . . . . 7 ⊢ (𝐴 ∈ No → Fun 𝐴)
11		funfn 6576	. . . . . . 7 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
12	10, 11	sylib 217	. . . . . 6 ⊢ (𝐴 ∈ No → 𝐴 Fn dom 𝐴)
13		fnsng 6598	. . . . . . 7 ⊢ ((dom 𝐴 ∈ On ∧ 𝑋 ∈ {1o, 2o}) → {⟨dom 𝐴, 𝑋⟩} Fn {dom 𝐴})
14	1, 2, 13	sylancl 587	. . . . . 6 ⊢ (𝐴 ∈ No → {⟨dom 𝐴, 𝑋⟩} Fn {dom 𝐴})
15		disjsn 4715	. . . . . . 7 ⊢ ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
16	7, 15	sylibr 233	. . . . . 6 ⊢ (𝐴 ∈ No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
17		snidg 4662	. . . . . . 7 ⊢ (dom 𝐴 ∈ On → dom 𝐴 ∈ {dom 𝐴})
18	1, 17	syl 17	. . . . . 6 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ {dom 𝐴})
19		fvun2 6981	. . . . . 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 1375	. . . . 5 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴) = ({⟨dom 𝐴, 𝑋⟩}‘dom 𝐴))
21		fvsng 7175	. . . . . 6 ⊢ ((dom 𝐴 ∈ On ∧ 𝑋 ∈ {1o, 2o}) → ({⟨dom 𝐴, 𝑋⟩}‘dom 𝐴) = 𝑋)
22	1, 2, 21	sylancl 587	. . . . 5 ⊢ (𝐴 ∈ No → ({⟨dom 𝐴, 𝑋⟩}‘dom 𝐴) = 𝑋)
23	20, 22	eqtrd 2773	. . . 4 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴) = 𝑋)
24	4, 9, 23	3netr4d 3019	. . 3 ⊢ (𝐴 ∈ No → (𝐴‘dom 𝐴) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴))
25		fveq2 6889	. . . . 5 ⊢ (𝑥 = dom 𝐴 → (𝐴‘𝑥) = (𝐴‘dom 𝐴))
26		fveq2 6889	. . . . 5 ⊢ (𝑥 = dom 𝐴 → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴))
27	25, 26	neeq12d 3003	. . . 4 ⊢ (𝑥 = dom 𝐴 → ((𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) ↔ (𝐴‘dom 𝐴) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴)))
28	27	onintss 6413	. . 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 6372	. . . . . . . 8 ⊢ (𝑦 ∈ On → Ord 𝑦)
31		ordtri2 6397	. . . . . . . . . 10 ⊢ ((Ord 𝑦 ∧ Ord dom 𝐴) → (𝑦 ∈ dom 𝐴 ↔ ¬ (𝑦 = dom 𝐴 ∨ dom 𝐴 ∈ 𝑦)))
32		eqcom 2740	. . . . . . . . . . . . 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 869	. . . . . . . . . . . 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 6391	. . . . . . . . . . 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 460	. . . . . . . . 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 598	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On) → (𝑦 ∈ dom 𝐴 ↔ ¬ dom 𝐴 ⊆ 𝑦))
43	12	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → 𝐴 Fn dom 𝐴)
44	14	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → {⟨dom 𝐴, 𝑋⟩} Fn {dom 𝐴})
45	16	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
46		simp3 1139	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → 𝑦 ∈ dom 𝐴)
47		fvun1 6980	. . . . . . . . . 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 1375	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) = (𝐴‘𝑦))
49	48	eqcomd 2739	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → (𝐴‘𝑦) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦))
50	49	3expia 1122	. . . . . . 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 2958	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On) → ((𝐴‘𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) → dom 𝐴 ⊆ 𝑦))
53	52	ralrimiva 3147	. . . 4 ⊢ (𝐴 ∈ No → ∀𝑦 ∈ On ((𝐴‘𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) → dom 𝐴 ⊆ 𝑦))
54		fveq2 6889	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴‘𝑥) = (𝐴‘𝑦))
55		fveq2 6889	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦))
56	54, 55	neeq12d 3003	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) ↔ (𝐴‘𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦)))
57	56	ralrab 3689	. . . 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 233	. . 3 ⊢ (𝐴 ∈ No → ∀𝑦 ∈ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)}dom 𝐴 ⊆ 𝑦)
59		ssint 4968	. . 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 233	. 2 ⊢ (𝐴 ∈ No → dom 𝐴 ⊆ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)})
61	29, 60	eqssd 3999	1 ⊢ (𝐴 ∈ No → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)} = dom 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  ∀wral 3062  {crab 3433   ∪ cun 3946   ∩ cin 3947   ⊆ wss 3948  ∅c0 4322  {csn 4628  {cpr 4630  ⟨cop 4634  ∩ cint 4950  dom cdm 5676  Ord word 6361  Oncon0 6362  Fun wfun 6535   Fn wfn 6536  ‘cfv 6541  1_oc1o 8456  2_oc2o 8457   No csur 27133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  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 6365  df-on 6366  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-1o 8463  df-2o 8464  df-no 27136

This theorem is referenced by:  noextendlt  27162  noextendgt  27163