Theorem nosepon 27866

Description: Given two unequal surreals, the minimal ordinal at which they differ is an ordinal. (Contributed by Scott Fenton, 21-Sep-2020.)

Assertion

Ref	Expression
nosepon	⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ On)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nosepon

Step	Hyp	Ref	Expression
1		df-ne 2962	. . . . . . . 8 ⊢ ((𝐴‘𝑥) ≠ (𝐵‘𝑥) ↔ ¬ (𝐴‘𝑥) = (𝐵‘𝑥))
2	1	rexbii 3115	. . . . . . 7 ⊢ (∃𝑥 ∈ On (𝐴‘𝑥) ≠ (𝐵‘𝑥) ↔ ∃𝑥 ∈ On ¬ (𝐴‘𝑥) = (𝐵‘𝑥))
3	2	notbii 323	. . . . . 6 ⊢ (¬ ∃𝑥 ∈ On (𝐴‘𝑥) ≠ (𝐵‘𝑥) ↔ ¬ ∃𝑥 ∈ On ¬ (𝐴‘𝑥) = (𝐵‘𝑥))
4		dfral2 3119	. . . . . 6 ⊢ (∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥) ↔ ¬ ∃𝑥 ∈ On ¬ (𝐴‘𝑥) = (𝐵‘𝑥))
5	3, 4	bitr4i 281	. . . . 5 ⊢ (¬ ∃𝑥 ∈ On (𝐴‘𝑥) ≠ (𝐵‘𝑥) ↔ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥))
6		nodmord 27854	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ No → Ord dom 𝐴)
7		nodmord 27854	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ No → Ord dom 𝐵)
8		ordtri3or 6400	. . . . . . . . . . . . 13 ⊢ ((Ord dom 𝐴 ∧ Ord dom 𝐵) → (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐴 = dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴))
9	6, 7, 8	syl2an 608	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐴 = dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴))
10		3orass 1106	. . . . . . . . . . . . 13 ⊢ ((dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐴 = dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴) ↔ (dom 𝐴 ∈ dom 𝐵 ∨ (dom 𝐴 = dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴)))
11		or12 934	. . . . . . . . . . . . 13 ⊢ ((dom 𝐴 ∈ dom 𝐵 ∨ (dom 𝐴 = dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴)) ↔ (dom 𝐴 = dom 𝐵 ∨ (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴)))
12	10, 11	bitri 278	. . . . . . . . . . . 12 ⊢ ((dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐴 = dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴) ↔ (dom 𝐴 = dom 𝐵 ∨ (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴)))
13	9, 12	sylib 221	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (dom 𝐴 = dom 𝐵 ∨ (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴)))
14	13	ord 878	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (¬ dom 𝐴 = dom 𝐵 → (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴)))
15		noseponlem 27865	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → ¬ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥))
16	15	3expia 1139	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (dom 𝐴 ∈ dom 𝐵 → ¬ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)))
17		noseponlem 27865	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ∧ dom 𝐵 ∈ dom 𝐴) → ¬ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐴‘𝑥))
18		eqcom 2773	. . . . . . . . . . . . . . 15 ⊢ ((𝐴‘𝑥) = (𝐵‘𝑥) ↔ (𝐵‘𝑥) = (𝐴‘𝑥))
19	18	ralbii 3114	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥) ↔ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐴‘𝑥))
20	17, 19	sylnibr 332	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ∧ dom 𝐵 ∈ dom 𝐴) → ¬ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥))
21	20	3expia 1139	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ) → (dom 𝐵 ∈ dom 𝐴 → ¬ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)))
22	21	ancoms 464	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (dom 𝐵 ∈ dom 𝐴 → ¬ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)))
23	16, 22	jaod 873	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴) → ¬ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)))
24	14, 23	syld 48	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (¬ dom 𝐴 = dom 𝐵 → ¬ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)))
25	24	con4d 116	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥) → dom 𝐴 = dom 𝐵))
26	25	3impia 1135	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)) → dom 𝐴 = dom 𝐵)
27		ordsson 7791	. . . . . . . . . 10 ⊢ (Ord dom 𝐴 → dom 𝐴 ⊆ On)
28		ssralv 4009	. . . . . . . . . 10 ⊢ (dom 𝐴 ⊆ On → (∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥) → ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥)))
29	6, 27, 28	3syl 19	. . . . . . . . 9 ⊢ (𝐴 ∈ No → (∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥) → ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥)))
30	29	adantr 486	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥) → ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥)))
31	30	3impia 1135	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)) → ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥))
32		nofun 27850	. . . . . . . . 9 ⊢ (𝐴 ∈ No → Fun 𝐴)
33	32	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)) → Fun 𝐴)
34		nofun 27850	. . . . . . . . 9 ⊢ (𝐵 ∈ No → Fun 𝐵)
35	34	3ad2ant2 1152	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)) → Fun 𝐵)
36		eqfunfv 7038	. . . . . . . 8 ⊢ ((Fun 𝐴 ∧ Fun 𝐵) → (𝐴 = 𝐵 ↔ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥))))
37	33, 35, 36	syl2anc 596	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)) → (𝐴 = 𝐵 ↔ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥))))
38	26, 31, 37	mpbir2and 726	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)) → 𝐴 = 𝐵)
39	38	3expia 1139	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥) → 𝐴 = 𝐵))
40	5, 39	biimtrid 245	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (¬ ∃𝑥 ∈ On (𝐴‘𝑥) ≠ (𝐵‘𝑥) → 𝐴 = 𝐵))
41	40	necon1ad 2978	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≠ 𝐵 → ∃𝑥 ∈ On (𝐴‘𝑥) ≠ (𝐵‘𝑥)))
42	41	3impia 1135	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → ∃𝑥 ∈ On (𝐴‘𝑥) ≠ (𝐵‘𝑥))
43		onintrab2 7805	. 2 ⊢ (∃𝑥 ∈ On (𝐴‘𝑥) ≠ (𝐵‘𝑥) ↔ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ On)
44	42, 43	sylib 221	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ On)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2961  ∀wral 3082  ∃wrex 3092  {crab 3419   ⊆ wss 3908  ∩ cint 4917  dom cdm 5666  Ord word 6366  Oncon0 6367  Fun wfun 6537  ‘cfv 6543   No csur 27841

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ord 6370  df-on 6371  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-1o 8462  df-2o 8463  df-no 27844

This theorem is used by:  nosepeq  27886  nosepssdm  27887  nodenselem4  27888  noresle  27898  nosupbnd2lem1  27916  noinfbnd2lem1  27931  noetasuplem4  27937  noetainflem4  27941