Theorem nosepon 33285

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 2988	. . . . . . . 8 ⊢ ((𝐴‘𝑥) ≠ (𝐵‘𝑥) ↔ ¬ (𝐴‘𝑥) = (𝐵‘𝑥))
2	1	rexbii 3210	. . . . . . 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 3200	. . . . . 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 33273	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ No → Ord dom 𝐴)
7		nodmord 33273	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ No → Ord dom 𝐵)
8		ordtri3or 6191	. . . . . . . . . . . . 13 ⊢ ((Ord dom 𝐴 ∧ Ord dom 𝐵) → (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐴 = dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴))
9	6, 7, 8	syl2an 598	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐴 = dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴))
10		3orass 1087	. . . . . . . . . . . . 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 918	. . . . . . . . . . . . 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 861	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (¬ dom 𝐴 = dom 𝐵 → (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴)))
15		noseponlem 33284	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → ¬ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥))
16	15	3expia 1118	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (dom 𝐴 ∈ dom 𝐵 → ¬ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)))
17		noseponlem 33284	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ∧ dom 𝐵 ∈ dom 𝐴) → ¬ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐴‘𝑥))
18		eqcom 2805	. . . . . . . . . . . . . . 15 ⊢ ((𝐴‘𝑥) = (𝐵‘𝑥) ↔ (𝐵‘𝑥) = (𝐴‘𝑥))
19	18	ralbii 3133	. . . . . . . . . . . . . 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 1118	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ) → (dom 𝐵 ∈ dom 𝐴 → ¬ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)))
22	21	ancoms 462	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (dom 𝐵 ∈ dom 𝐴 → ¬ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)))
23	16, 22	jaod 856	. . . . . . . . . 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 1114	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)) → dom 𝐴 = dom 𝐵)
27		ordsson 7484	. . . . . . . . . 10 ⊢ (Ord dom 𝐴 → dom 𝐴 ⊆ On)
28		ssralv 3981	. . . . . . . . . 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 484	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥) → ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥)))
31	30	3impia 1114	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)) → ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥))
32		nofun 33269	. . . . . . . . 9 ⊢ (𝐴 ∈ No → Fun 𝐴)
33	32	3ad2ant1 1130	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)) → Fun 𝐴)
34		nofun 33269	. . . . . . . . 9 ⊢ (𝐵 ∈ No → Fun 𝐵)
35	34	3ad2ant2 1131	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)) → Fun 𝐵)
36		eqfunfv 6784	. . . . . . . 8 ⊢ ((Fun 𝐴 ∧ Fun 𝐵) → (𝐴 = 𝐵 ↔ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥))))
37	33, 35, 36	syl2anc 587	. . . . . . 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 712	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)) → 𝐴 = 𝐵)
39	38	3expia 1118	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥) → 𝐴 = 𝐵))
40	5, 39	syl5bi 245	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (¬ ∃𝑥 ∈ On (𝐴‘𝑥) ≠ (𝐵‘𝑥) → 𝐴 = 𝐵))
41	40	necon1ad 3004	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≠ 𝐵 → ∃𝑥 ∈ On (𝐴‘𝑥) ≠ (𝐵‘𝑥)))
42	41	3impia 1114	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → ∃𝑥 ∈ On (𝐴‘𝑥) ≠ (𝐵‘𝑥))
43		onintrab2 7497	. 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 399   ∨ wo 844   ∨ w3o 1083   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  {crab 3110   ⊆ wss 3881  ∩ cint 4838  dom cdm 5519  Ord word 6158  Oncon0 6159  Fun wfun 6318  ‘cfv 6324   No csur 33260

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-1o 8085  df-2o 8086  df-no 33263

This theorem is referenced by:  nosepeq  33302  nosepssdm  33303  nodenselem4  33304  noresle  33313  nosupbnd2lem1  33328  noetalem3  33332