Theorem nosepon 33795

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 2943	. . . . . . . 8 ⊢ ((𝐴‘𝑥) ≠ (𝐵‘𝑥) ↔ ¬ (𝐴‘𝑥) = (𝐵‘𝑥))
2	1	rexbii 3177	. . . . . . 7 ⊢ (∃𝑥 ∈ On (𝐴‘𝑥) ≠ (𝐵‘𝑥) ↔ ∃𝑥 ∈ On ¬ (𝐴‘𝑥) = (𝐵‘𝑥))
3	2	notbii 319	. . . . . 6 ⊢ (¬ ∃𝑥 ∈ On (𝐴‘𝑥) ≠ (𝐵‘𝑥) ↔ ¬ ∃𝑥 ∈ On ¬ (𝐴‘𝑥) = (𝐵‘𝑥))
4		dfral2 3164	. . . . . 6 ⊢ (∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥) ↔ ¬ ∃𝑥 ∈ On ¬ (𝐴‘𝑥) = (𝐵‘𝑥))
5	3, 4	bitr4i 277	. . . . 5 ⊢ (¬ ∃𝑥 ∈ On (𝐴‘𝑥) ≠ (𝐵‘𝑥) ↔ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥))
6		nodmord 33783	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ No → Ord dom 𝐴)
7		nodmord 33783	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ No → Ord dom 𝐵)
8		ordtri3or 6283	. . . . . . . . . . . . 13 ⊢ ((Ord dom 𝐴 ∧ Ord dom 𝐵) → (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐴 = dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴))
9	6, 7, 8	syl2an 595	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐴 = dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴))
10		3orass 1088	. . . . . . . . . . . . 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 917	. . . . . . . . . . . . 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 274	. . . . . . . . . . . 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 217	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (dom 𝐴 = dom 𝐵 ∨ (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴)))
14	13	ord 860	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (¬ dom 𝐴 = dom 𝐵 → (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴)))
15		noseponlem 33794	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → ¬ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥))
16	15	3expia 1119	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (dom 𝐴 ∈ dom 𝐵 → ¬ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)))
17		noseponlem 33794	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ∧ dom 𝐵 ∈ dom 𝐴) → ¬ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐴‘𝑥))
18		eqcom 2745	. . . . . . . . . . . . . . 15 ⊢ ((𝐴‘𝑥) = (𝐵‘𝑥) ↔ (𝐵‘𝑥) = (𝐴‘𝑥))
19	18	ralbii 3090	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥) ↔ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐴‘𝑥))
20	17, 19	sylnibr 328	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ∧ dom 𝐵 ∈ dom 𝐴) → ¬ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥))
21	20	3expia 1119	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ) → (dom 𝐵 ∈ dom 𝐴 → ¬ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)))
22	21	ancoms 458	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (dom 𝐵 ∈ dom 𝐴 → ¬ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)))
23	16, 22	jaod 855	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴) → ¬ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)))
24	14, 23	syld 47	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (¬ dom 𝐴 = dom 𝐵 → ¬ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)))
25	24	con4d 115	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥) → dom 𝐴 = dom 𝐵))
26	25	3impia 1115	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)) → dom 𝐴 = dom 𝐵)
27		ordsson 7610	. . . . . . . . . 10 ⊢ (Ord dom 𝐴 → dom 𝐴 ⊆ On)
28		ssralv 3983	. . . . . . . . . 10 ⊢ (dom 𝐴 ⊆ On → (∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥) → ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥)))
29	6, 27, 28	3syl 18	. . . . . . . . 9 ⊢ (𝐴 ∈ No → (∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥) → ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥)))
30	29	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥) → ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥)))
31	30	3impia 1115	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)) → ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥))
32		nofun 33779	. . . . . . . . 9 ⊢ (𝐴 ∈ No → Fun 𝐴)
33	32	3ad2ant1 1131	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)) → Fun 𝐴)
34		nofun 33779	. . . . . . . . 9 ⊢ (𝐵 ∈ No → Fun 𝐵)
35	34	3ad2ant2 1132	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)) → Fun 𝐵)
36		eqfunfv 6896	. . . . . . . 8 ⊢ ((Fun 𝐴 ∧ Fun 𝐵) → (𝐴 = 𝐵 ↔ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥))))
37	33, 35, 36	syl2anc 583	. . . . . . 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 709	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)) → 𝐴 = 𝐵)
39	38	3expia 1119	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥) → 𝐴 = 𝐵))
40	5, 39	syl5bi 241	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (¬ ∃𝑥 ∈ On (𝐴‘𝑥) ≠ (𝐵‘𝑥) → 𝐴 = 𝐵))
41	40	necon1ad 2959	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≠ 𝐵 → ∃𝑥 ∈ On (𝐴‘𝑥) ≠ (𝐵‘𝑥)))
42	41	3impia 1115	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → ∃𝑥 ∈ On (𝐴‘𝑥) ≠ (𝐵‘𝑥))
43		onintrab2 7624	. 2 ⊢ (∃𝑥 ∈ On (𝐴‘𝑥) ≠ (𝐵‘𝑥) ↔ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ On)
44	42, 43	sylib 217	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ On)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   ∨ w3o 1084   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  {crab 3067   ⊆ wss 3883  ∩ cint 4876  dom cdm 5580  Ord word 6250  Oncon0 6251  Fun wfun 6412  ‘cfv 6418   No csur 33770

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-1o 8267  df-2o 8268  df-no 33773

This theorem is referenced by:  nosepeq  33815  nosepssdm  33816  nodenselem4  33817  noresle  33827  nosupbnd2lem1  33845  noinfbnd2lem1  33860  noetasuplem4  33866  noetainflem4  33870