Theorem noextenddif 27719

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 27701	. . 3 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
2		noextend.1	. . . . . 6 ⊢ 𝑋 ∈ {1o, 2o}
3	2	nosgnn0i 27710	. . . . 5 ⊢ ∅ ≠ 𝑋
4	3	a1i 11	. . . 4 ⊢ (𝐴 ∈ No → ∅ ≠ 𝑋)
5		nodmord 27704	. . . . . 6 ⊢ (𝐴 ∈ No → Ord dom 𝐴)
6		ordirr 6358	. . . . . 6 ⊢ (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
7	5, 6	syl 17	. . . . 5 ⊢ (𝐴 ∈ No → ¬ dom 𝐴 ∈ dom 𝐴)
8		ndmfv 6893	. . . . 5 ⊢ (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
9	7, 8	syl 17	. . . 4 ⊢ (𝐴 ∈ No → (𝐴‘dom 𝐴) = ∅)
10		nofun 27700	. . . . . . 7 ⊢ (𝐴 ∈ No → Fun 𝐴)
11		funfn 6545	. . . . . . 7 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
12	10, 11	sylib 220	. . . . . 6 ⊢ (𝐴 ∈ No → 𝐴 Fn dom 𝐴)
13		fnsng 6567	. . . . . . 7 ⊢ ((dom 𝐴 ∈ On ∧ 𝑋 ∈ {1o, 2o}) → {⟨dom 𝐴, 𝑋⟩} Fn {dom 𝐴})
14	1, 2, 13	sylancl 595	. . . . . 6 ⊢ (𝐴 ∈ No → {⟨dom 𝐴, 𝑋⟩} Fn {dom 𝐴})
15		disjsn 4667	. . . . . . 7 ⊢ ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
16	7, 15	sylibr 236	. . . . . 6 ⊢ (𝐴 ∈ No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
17		snidg 4616	. . . . . . 7 ⊢ (dom 𝐴 ∈ On → dom 𝐴 ∈ {dom 𝐴})
18	1, 17	syl 17	. . . . . 6 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ {dom 𝐴})
19		fvun2 6953	. . . . . 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 1392	. . . . 5 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴) = ({⟨dom 𝐴, 𝑋⟩}‘dom 𝐴))
21		fvsng 7158	. . . . . 6 ⊢ ((dom 𝐴 ∈ On ∧ 𝑋 ∈ {1o, 2o}) → ({⟨dom 𝐴, 𝑋⟩}‘dom 𝐴) = 𝑋)
22	1, 2, 21	sylancl 595	. . . . 5 ⊢ (𝐴 ∈ No → ({⟨dom 𝐴, 𝑋⟩}‘dom 𝐴) = 𝑋)
23	20, 22	eqtrd 2796	. . . 4 ⊢ (𝐴 ∈ No → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴) = 𝑋)
24	4, 9, 23	3netr4d 3033	. . 3 ⊢ (𝐴 ∈ No → (𝐴‘dom 𝐴) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴))
25		fveq2 6861	. . . . 5 ⊢ (𝑥 = dom 𝐴 → (𝐴‘𝑥) = (𝐴‘dom 𝐴))
26		fveq2 6861	. . . . 5 ⊢ (𝑥 = dom 𝐴 → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴))
27	25, 26	neeq12d 3017	. . . 4 ⊢ (𝑥 = dom 𝐴 → ((𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) ↔ (𝐴‘dom 𝐴) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘dom 𝐴)))
28	27	onintss 6392	. . 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 6350	. . . . . . . 8 ⊢ (𝑦 ∈ On → Ord 𝑦)
31		ordtri2 6375	. . . . . . . . . 10 ⊢ ((Ord 𝑦 ∧ Ord dom 𝐴) → (𝑦 ∈ dom 𝐴 ↔ ¬ (𝑦 = dom 𝐴 ∨ dom 𝐴 ∈ 𝑦)))
32		eqcom 2768	. . . . . . . . . . . . 13 ⊢ (𝑦 = dom 𝐴 ↔ dom 𝐴 = 𝑦)
33	32	orbi1i 924	. . . . . . . . . . . 12 ⊢ ((𝑦 = dom 𝐴 ∨ dom 𝐴 ∈ 𝑦) ↔ (dom 𝐴 = 𝑦 ∨ dom 𝐴 ∈ 𝑦))
34		orcom 881	. . . . . . . . . . . 12 ⊢ ((dom 𝐴 = 𝑦 ∨ dom 𝐴 ∈ 𝑦) ↔ (dom 𝐴 ∈ 𝑦 ∨ dom 𝐴 = 𝑦))
35	33, 34	bitri 277	. . . . . . . . . . 11 ⊢ ((𝑦 = dom 𝐴 ∨ dom 𝐴 ∈ 𝑦) ↔ (dom 𝐴 ∈ 𝑦 ∨ dom 𝐴 = 𝑦))
36	35	notbii 322	. . . . . . . . . 10 ⊢ (¬ (𝑦 = dom 𝐴 ∨ dom 𝐴 ∈ 𝑦) ↔ ¬ (dom 𝐴 ∈ 𝑦 ∨ dom 𝐴 = 𝑦))
37	31, 36	bitrdi 289	. . . . . . . . 9 ⊢ ((Ord 𝑦 ∧ Ord dom 𝐴) → (𝑦 ∈ dom 𝐴 ↔ ¬ (dom 𝐴 ∈ 𝑦 ∨ dom 𝐴 = 𝑦)))
38		ordsseleq 6369	. . . . . . . . . . 11 ⊢ ((Ord dom 𝐴 ∧ Ord 𝑦) → (dom 𝐴 ⊆ 𝑦 ↔ (dom 𝐴 ∈ 𝑦 ∨ dom 𝐴 = 𝑦)))
39	38	notbid 320	. . . . . . . . . 10 ⊢ ((Ord dom 𝐴 ∧ Ord 𝑦) → (¬ dom 𝐴 ⊆ 𝑦 ↔ ¬ (dom 𝐴 ∈ 𝑦 ∨ dom 𝐴 = 𝑦)))
40	39	ancoms 462	. . . . . . . . 9 ⊢ ((Ord 𝑦 ∧ Ord dom 𝐴) → (¬ dom 𝐴 ⊆ 𝑦 ↔ ¬ (dom 𝐴 ∈ 𝑦 ∨ dom 𝐴 = 𝑦)))
41	37, 40	bitr4d 284	. . . . . . . 8 ⊢ ((Ord 𝑦 ∧ Ord dom 𝐴) → (𝑦 ∈ dom 𝐴 ↔ ¬ dom 𝐴 ⊆ 𝑦))
42	30, 5, 41	syl2anr 606	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On) → (𝑦 ∈ dom 𝐴 ↔ ¬ dom 𝐴 ⊆ 𝑦))
43	12	3ad2ant1 1145	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → 𝐴 Fn dom 𝐴)
44	14	3ad2ant1 1145	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → {⟨dom 𝐴, 𝑋⟩} Fn {dom 𝐴})
45	16	3ad2ant1 1145	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
46		simp3 1150	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → 𝑦 ∈ dom 𝐴)
47		fvun1 6952	. . . . . . . . . 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 1392	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) = (𝐴‘𝑦))
49	48	eqcomd 2767	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On ∧ 𝑦 ∈ dom 𝐴) → (𝐴‘𝑦) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦))
50	49	3expia 1133	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On) → (𝑦 ∈ dom 𝐴 → (𝐴‘𝑦) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦)))
51	42, 50	sylbird 262	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On) → (¬ dom 𝐴 ⊆ 𝑦 → (𝐴‘𝑦) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦)))
52	51	necon1ad 2973	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ On) → ((𝐴‘𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) → dom 𝐴 ⊆ 𝑦))
53	52	ralrimiva 3153	. . . 4 ⊢ (𝐴 ∈ No → ∀𝑦 ∈ On ((𝐴‘𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦) → dom 𝐴 ⊆ 𝑦))
54		fveq2 6861	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴‘𝑥) = (𝐴‘𝑦))
55		fveq2 6861	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) = ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦))
56	54, 55	neeq12d 3017	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥) ↔ (𝐴‘𝑦) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑦)))
57	56	ralrab 3655	. . . 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 236	. . 3 ⊢ (𝐴 ∈ No → ∀𝑦 ∈ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)}dom 𝐴 ⊆ 𝑦)
59		ssint 4919	. . 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 236	. 2 ⊢ (𝐴 ∈ No → dom 𝐴 ⊆ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)})
61	29, 60	eqssd 3951	1 ⊢ (𝐴 ∈ No → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)} = dom 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  ∀wral 3075  {crab 3413   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  ∅c0 4283  {csn 4579  {cpr 4581  ⟨cop 4585  ∩ cint 4902  dom cdm 5643  Ord word 6339  Oncon0 6340  Fun wfun 6509   Fn wfn 6510  ‘cfv 6515  1_oc1o 8423  2_oc2o 8424   No csur 27691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-br 5098  df-opab 5160  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-ord 6343  df-on 6344  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-fv 6523  df-1o 8430  df-2o 8431  df-no 27694

This theorem is referenced by:  noextendlt  27720  noextendgt  27721