Theorem noextenddif 31821

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 31803	. . 3 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
2		noextend.1	. . . . . 6 ⊢ 𝑋 ∈ {1𝑜, 2𝑜}
3	2	nosgnn0i 31812	. . . . 5 ⊢ ∅ ≠ 𝑋
4	3	a1i 11	. . . 4 ⊢ (𝐴 ∈ No → ∅ ≠ 𝑋)
5		nodmord 31806	. . . . . 6 ⊢ (𝐴 ∈ No → Ord dom 𝐴)
6		ordirr 5741	. . . . . 6 ⊢ (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
7	5, 6	syl 17	. . . . 5 ⊢ (𝐴 ∈ No → ¬ dom 𝐴 ∈ dom 𝐴)
8		ndmfv 6218	. . . . 5 ⊢ (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
9	7, 8	syl 17	. . . 4 ⊢ (𝐴 ∈ No → (𝐴‘dom 𝐴) = ∅)
10		nofun 31802	. . . . . . 7 ⊢ (𝐴 ∈ No → Fun 𝐴)
11		funfn 5918	. . . . . . 7 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
12	10, 11	sylib 208	. . . . . 6 ⊢ (𝐴 ∈ No → 𝐴 Fn dom 𝐴)
13		fnsng 5938	. . . . . . 7 ⊢ ((dom 𝐴 ∈ On ∧ 𝑋 ∈ {1𝑜, 2𝑜}) → {⟨dom 𝐴, 𝑋⟩} Fn {dom 𝐴})
14	1, 2, 13	sylancl 694	. . . . . 6 ⊢ (𝐴 ∈ No → {⟨dom 𝐴, 𝑋⟩} Fn {dom 𝐴})
15		disjsn 4246	. . . . . . 7 ⊢ ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
16	7, 15	sylibr 224	. . . . . 6 ⊢ (𝐴 ∈ No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
17		snidg 4206	. . . . . . 7 ⊢ (dom 𝐴 ∈ On → dom 𝐴 ∈ {dom 𝐴})
18	1, 17	syl 17	. . . . . 6 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ {dom 𝐴})
19		fvun2 6270	. . . . . 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 1330	. . . . 5 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴) = ({⟨dom 𝐴, 𝑋⟩}‘dom 𝐴))
21		fvsng 6447	. . . . . 6 ⊢ ((dom 𝐴 ∈ On ∧ 𝑋 ∈ {1𝑜, 2𝑜}) → ({⟨dom 𝐴, 𝑋⟩}‘dom 𝐴) = 𝑋)
22	1, 2, 21	sylancl 694	. . . . 5 ⊢ (𝐴 ∈ No → ({⟨dom 𝐴, 𝑋⟩}‘dom 𝐴) = 𝑋)
23	20, 22	eqtrd 2656	. . . 4 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴) = 𝑋)
24	4, 9, 23	3netr4d 2871	. . 3 ⊢ (𝐴 ∈ No → (𝐴‘dom 𝐴) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴))
25		fveq2 6191	. . . . 5 ⊢ (𝑥 = dom 𝐴 → (𝐴‘𝑥) = (𝐴‘dom 𝐴))
26		fveq2 6191	. . . . 5 ⊢ (𝑥 = dom 𝐴 → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴))
27	25, 26	neeq12d 2855	. . . 4 ⊢ (𝑥 = dom 𝐴 → ((𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) ↔ (𝐴‘dom 𝐴) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴)))
28	27	onintss 5775	. . 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 5733	. . . . . . . 8 ⊢ (𝑦 ∈ On → Ord 𝑦)
31		ordtri2 5758	. . . . . . . . . 10 ⊢ ((Ord 𝑦 ∧ Ord dom 𝐴) → (𝑦 ∈ dom 𝐴 ↔ ¬ (𝑦 = dom 𝐴 ∨ dom 𝐴 ∈ 𝑦)))
32		eqcom 2629	. . . . . . . . . . . . 13 ⊢ (𝑦 = dom 𝐴 ↔ dom 𝐴 = 𝑦)
33	32	orbi1i 542	. . . . . . . . . . . 12 ⊢ ((𝑦 = dom 𝐴 ∨ dom 𝐴 ∈ 𝑦) ↔ (dom 𝐴 = 𝑦 ∨ dom 𝐴 ∈ 𝑦))
34		orcom 402	. . . . . . . . . . . 12 ⊢ ((dom 𝐴 = 𝑦 ∨ dom 𝐴 ∈ 𝑦) ↔ (dom 𝐴 ∈ 𝑦 ∨ dom 𝐴 = 𝑦))
35	33, 34	bitri 264	. . . . . . . . . . 11 ⊢ ((𝑦 = dom 𝐴 ∨ dom 𝐴 ∈ 𝑦) ↔ (dom 𝐴 ∈ 𝑦 ∨ dom 𝐴 = 𝑦))
36	35	notbii 310	. . . . . . . . . 10 ⊢ (¬ (𝑦 = dom 𝐴 ∨ dom 𝐴 ∈ 𝑦) ↔ ¬ (dom 𝐴 ∈ 𝑦 ∨ dom 𝐴 = 𝑦))
37	31, 36	syl6bb 276	. . . . . . . . 9 ⊢ ((Ord 𝑦 ∧ Ord dom 𝐴) → (𝑦 ∈ dom 𝐴 ↔ ¬ (dom 𝐴 ∈ 𝑦 ∨ dom 𝐴 = 𝑦)))
38		ordsseleq 5752	. . . . . . . . . . 11 ⊢ ((Ord dom 𝐴 ∧ Ord 𝑦) → (dom 𝐴 ⊆ 𝑦 ↔ (dom 𝐴 ∈ 𝑦 ∨ dom 𝐴 = 𝑦)))
39	38	notbid 308	. . . . . . . . . 10 ⊢ ((Ord dom 𝐴 ∧ Ord 𝑦) → (¬ dom 𝐴 ⊆ 𝑦 ↔ ¬ (dom 𝐴 ∈ 𝑦 ∨ dom 𝐴 = 𝑦)))
40	39	ancoms 469	. . . . . . . . 9 ⊢ ((Ord 𝑦 ∧ Ord dom 𝐴) → (¬ dom 𝐴 ⊆ 𝑦 ↔ ¬ (dom 𝐴 ∈ 𝑦 ∨ dom 𝐴 = 𝑦)))
41	37, 40	bitr4d 271	. . . . . . . 8 ⊢ ((Ord 𝑦 ∧ Ord dom 𝐴) → (𝑦 ∈ dom 𝐴 ↔ ¬ dom 𝐴 ⊆ 𝑦))
42	30, 5, 41	syl2anr 495	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On) → (𝑦 ∈ dom 𝐴 ↔ ¬ dom 𝐴 ⊆ 𝑦))
43	12	3ad2ant1 1082	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → 𝐴 Fn dom 𝐴)
44	14	3ad2ant1 1082	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → {⟨dom 𝐴, 𝑋⟩} Fn {dom 𝐴})
45	16	3ad2ant1 1082	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
46		simp3 1063	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → 𝑦 ∈ dom 𝐴)
47		fvun1 6269	. . . . . . . . . 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 1330	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) = (𝐴‘𝑦))
49	48	eqcomd 2628	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → (𝐴‘𝑦) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦))
50	49	3expia 1267	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On) → (𝑦 ∈ dom 𝐴 → (𝐴‘𝑦) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦)))
51	42, 50	sylbird 250	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On) → (¬ dom 𝐴 ⊆ 𝑦 → (𝐴‘𝑦) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦)))
52	51	necon1ad 2811	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On) → ((𝐴‘𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) → dom 𝐴 ⊆ 𝑦))
53	52	ralrimiva 2966	. . . 4 ⊢ (𝐴 ∈ No → ∀𝑦 ∈ On ((𝐴‘𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) → dom 𝐴 ⊆ 𝑦))
54		fveq2 6191	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴‘𝑥) = (𝐴‘𝑦))
55		fveq2 6191	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦))
56	54, 55	neeq12d 2855	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) ↔ (𝐴‘𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦)))
57	56	ralrab 3368	. . . 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 224	. . 3 ⊢ (𝐴 ∈ No → ∀𝑦 ∈ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)}dom 𝐴 ⊆ 𝑦)
59		ssint 4493	. . 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 224	. 2 ⊢ (𝐴 ∈ No → dom 𝐴 ⊆ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)})
61	29, 60	eqssd 3620	1 ⊢ (𝐴 ∈ No → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)} = dom 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  {crab 2916   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {csn 4177  {cpr 4179  ⟨cop 4183  ∩ cint 4475  dom cdm 5114  Ord word 5722  Oncon0 5723  Fun wfun 5882   Fn wfn 5883  ‘cfv 5888  1_𝑜c1o 7553  2_𝑜c2o 7554   No csur 31793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1o 7560  df-2o 7561  df-no 31796

This theorem is referenced by:  noextendlt  31822  noextendgt  31823