Theorem nosepon 27810

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 2959	. . . . . . . 8 ⊢ ((𝐴‘𝑥) ≠ (𝐵‘𝑥) ↔ ¬ (𝐴‘𝑥) = (𝐵‘𝑥))
2	1	rexbii 3112	. . . . . . 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 3116	. . . . . 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 27798	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ No → Ord dom 𝐴)
7		nodmord 27798	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ No → Ord dom 𝐵)
8		ordtri3or 6395	. . . . . . . . . . . . 13 ⊢ ((Ord dom 𝐴 ∧ Ord dom 𝐵) → (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐴 = dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴))
9	6, 7, 8	syl2an 607	. . . . . . . . . . . 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 933	. . . . . . . . . . . . 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 877	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (¬ dom 𝐴 = dom 𝐵 → (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴)))
15		noseponlem 27809	. . . . . . . . . . . 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 27809	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ∧ dom 𝐵 ∈ dom 𝐴) → ¬ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐴‘𝑥))
18		eqcom 2770	. . . . . . . . . . . . . . 15 ⊢ ((𝐴‘𝑥) = (𝐵‘𝑥) ↔ (𝐵‘𝑥) = (𝐴‘𝑥))
19	18	ralbii 3111	. . . . . . . . . . . . . 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 463	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (dom 𝐵 ∈ dom 𝐴 → ¬ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)))
23	16, 22	jaod 872	. . . . . . . . . 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 7783	. . . . . . . . . 10 ⊢ (Ord dom 𝐴 → dom 𝐴 ⊆ On)
28		ssralv 4007	. . . . . . . . . 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 485	. . . . . . . 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 27794	. . . . . . . . 9 ⊢ (𝐴 ∈ No → Fun 𝐴)
33	32	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)) → Fun 𝐴)
34		nofun 27794	. . . . . . . . 9 ⊢ (𝐵 ∈ No → Fun 𝐵)
35	34	3ad2ant2 1152	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)) → Fun 𝐵)
36		eqfunfv 7033	. . . . . . . 8 ⊢ ((Fun 𝐴 ∧ Fun 𝐵) → (𝐴 = 𝐵 ↔ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥))))
37	33, 35, 36	syl2anc 595	. . . . . . 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 725	. . . . . 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 2975	. . 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 7797	. 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)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∨ w3o 1102   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  {crab 3416   ⊆ wss 3906  ∩ cint 4913  dom cdm 5663  Ord word 6361  Oncon0 6362  Fun wfun 6532  ‘cfv 6538   No csur 27785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-1o 8454  df-2o 8455  df-no 27788

This theorem is referenced by:  nosepeq  27830  nosepssdm  27831  nodenselem4  27832  noresle  27842  nosupbnd2lem1  27860  noinfbnd2lem1  27875  noetasuplem4  27881  noetainflem4  27885