Theorem noextenddif 27016

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 26998	. . 3 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
2		noextend.1	. . . . . 6 ⊢ 𝑋 ∈ {1o, 2o}
3	2	nosgnn0i 27007	. . . . 5 ⊢ ∅ ≠ 𝑋
4	3	a1i 11	. . . 4 ⊢ (𝐴 ∈ No → ∅ ≠ 𝑋)
5		nodmord 27001	. . . . . 6 ⊢ (𝐴 ∈ No → Ord dom 𝐴)
6		ordirr 6335	. . . . . 6 ⊢ (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
7	5, 6	syl 17	. . . . 5 ⊢ (𝐴 ∈ No → ¬ dom 𝐴 ∈ dom 𝐴)
8		ndmfv 6877	. . . . 5 ⊢ (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
9	7, 8	syl 17	. . . 4 ⊢ (𝐴 ∈ No → (𝐴‘dom 𝐴) = ∅)
10		nofun 26997	. . . . . . 7 ⊢ (𝐴 ∈ No → Fun 𝐴)
11		funfn 6531	. . . . . . 7 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
12	10, 11	sylib 217	. . . . . 6 ⊢ (𝐴 ∈ No → 𝐴 Fn dom 𝐴)
13		fnsng 6553	. . . . . . 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 4672	. . . . . . 7 ⊢ ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
16	7, 15	sylibr 233	. . . . . 6 ⊢ (𝐴 ∈ No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
17		snidg 4620	. . . . . . 7 ⊢ (dom 𝐴 ∈ On → dom 𝐴 ∈ {dom 𝐴})
18	1, 17	syl 17	. . . . . 6 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ {dom 𝐴})
19		fvun2 6933	. . . . . 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 1374	. . . . 5 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴) = ({⟨dom 𝐴, 𝑋⟩}‘dom 𝐴))
21		fvsng 7126	. . . . . 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 2776	. . . 4 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴) = 𝑋)
24	4, 9, 23	3netr4d 3021	. . 3 ⊢ (𝐴 ∈ No → (𝐴‘dom 𝐴) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴))
25		fveq2 6842	. . . . 5 ⊢ (𝑥 = dom 𝐴 → (𝐴‘𝑥) = (𝐴‘dom 𝐴))
26		fveq2 6842	. . . . 5 ⊢ (𝑥 = dom 𝐴 → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴))
27	25, 26	neeq12d 3005	. . . 4 ⊢ (𝑥 = dom 𝐴 → ((𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) ↔ (𝐴‘dom 𝐴) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴)))
28	27	onintss 6368	. . 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 6327	. . . . . . . 8 ⊢ (𝑦 ∈ On → Ord 𝑦)
31		ordtri2 6352	. . . . . . . . . 10 ⊢ ((Ord 𝑦 ∧ Ord dom 𝐴) → (𝑦 ∈ dom 𝐴 ↔ ¬ (𝑦 = dom 𝐴 ∨ dom 𝐴 ∈ 𝑦)))
32		eqcom 2743	. . . . . . . . . . . . 13 ⊢ (𝑦 = dom 𝐴 ↔ dom 𝐴 = 𝑦)
33	32	orbi1i 912	. . . . . . . . . . . 12 ⊢ ((𝑦 = dom 𝐴 ∨ dom 𝐴 ∈ 𝑦) ↔ (dom 𝐴 = 𝑦 ∨ dom 𝐴 ∈ 𝑦))
34		orcom 868	. . . . . . . . . . . 12 ⊢ ((dom 𝐴 = 𝑦 ∨ dom 𝐴 ∈ 𝑦) ↔ (dom 𝐴 ∈ 𝑦 ∨ dom 𝐴 = 𝑦))
35	33, 34	bitri 274	. . . . . . . . . . 11 ⊢ ((𝑦 = dom 𝐴 ∨ dom 𝐴 ∈ 𝑦) ↔ (dom 𝐴 ∈ 𝑦 ∨ dom 𝐴 = 𝑦))
36	35	notbii 319	. . . . . . . . . 10 ⊢ (¬ (𝑦 = dom 𝐴 ∨ dom 𝐴 ∈ 𝑦) ↔ ¬ (dom 𝐴 ∈ 𝑦 ∨ dom 𝐴 = 𝑦))
37	31, 36	bitrdi 286	. . . . . . . . 9 ⊢ ((Ord 𝑦 ∧ Ord dom 𝐴) → (𝑦 ∈ dom 𝐴 ↔ ¬ (dom 𝐴 ∈ 𝑦 ∨ dom 𝐴 = 𝑦)))
38		ordsseleq 6346	. . . . . . . . . . 11 ⊢ ((Ord dom 𝐴 ∧ Ord 𝑦) → (dom 𝐴 ⊆ 𝑦 ↔ (dom 𝐴 ∈ 𝑦 ∨ dom 𝐴 = 𝑦)))
39	38	notbid 317	. . . . . . . . . 10 ⊢ ((Ord dom 𝐴 ∧ Ord 𝑦) → (¬ dom 𝐴 ⊆ 𝑦 ↔ ¬ (dom 𝐴 ∈ 𝑦 ∨ dom 𝐴 = 𝑦)))
40	39	ancoms 459	. . . . . . . . 9 ⊢ ((Ord 𝑦 ∧ Ord dom 𝐴) → (¬ dom 𝐴 ⊆ 𝑦 ↔ ¬ (dom 𝐴 ∈ 𝑦 ∨ dom 𝐴 = 𝑦)))
41	37, 40	bitr4d 281	. . . . . . . 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 6932	. . . . . . . . . 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 1374	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) = (𝐴‘𝑦))
49	48	eqcomd 2742	. . . . . . . 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 259	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On) → (¬ dom 𝐴 ⊆ 𝑦 → (𝐴‘𝑦) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦)))
52	51	necon1ad 2960	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On) → ((𝐴‘𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) → dom 𝐴 ⊆ 𝑦))
53	52	ralrimiva 3143	. . . 4 ⊢ (𝐴 ∈ No → ∀𝑦 ∈ On ((𝐴‘𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) → dom 𝐴 ⊆ 𝑦))
54		fveq2 6842	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴‘𝑥) = (𝐴‘𝑦))
55		fveq2 6842	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦))
56	54, 55	neeq12d 3005	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) ↔ (𝐴‘𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦)))
57	56	ralrab 3651	. . . 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 4925	. . 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 3961	1 ⊢ (𝐴 ∈ No → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)} = dom 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  {crab 3407   ∪ cun 3908   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282  {csn 4586  {cpr 4588  ⟨cop 4592  ∩ cint 4907  dom cdm 5633  Ord word 6316  Oncon0 6317  Fun wfun 6490   Fn wfn 6491  ‘cfv 6496  1_oc1o 8405  2_oc2o 8406   No csur 26988

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-1o 8412  df-2o 8413  df-no 26991

This theorem is referenced by:  noextendlt  27017  noextendgt  27018