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Theorem nosepon 27710
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 𝐴𝐵) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ On)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nosepon
StepHypRef Expression
1 df-ne 2941 . . . . . . . 8 ((𝐴𝑥) ≠ (𝐵𝑥) ↔ ¬ (𝐴𝑥) = (𝐵𝑥))
21rexbii 3094 . . . . . . 7 (∃𝑥 ∈ On (𝐴𝑥) ≠ (𝐵𝑥) ↔ ∃𝑥 ∈ On ¬ (𝐴𝑥) = (𝐵𝑥))
32notbii 320 . . . . . 6 (¬ ∃𝑥 ∈ On (𝐴𝑥) ≠ (𝐵𝑥) ↔ ¬ ∃𝑥 ∈ On ¬ (𝐴𝑥) = (𝐵𝑥))
4 dfral2 3099 . . . . . 6 (∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥) ↔ ¬ ∃𝑥 ∈ On ¬ (𝐴𝑥) = (𝐵𝑥))
53, 4bitr4i 278 . . . . 5 (¬ ∃𝑥 ∈ On (𝐴𝑥) ≠ (𝐵𝑥) ↔ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥))
6 nodmord 27698 . . . . . . . . . . . . 13 (𝐴 No → Ord dom 𝐴)
7 nodmord 27698 . . . . . . . . . . . . 13 (𝐵 No → Ord dom 𝐵)
8 ordtri3or 6416 . . . . . . . . . . . . 13 ((Ord dom 𝐴 ∧ Ord dom 𝐵) → (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐴 = dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴))
96, 7, 8syl2an 596 . . . . . . . . . . . 12 ((𝐴 No 𝐵 No ) → (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐴 = dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴))
10 3orass 1090 . . . . . . . . . . . . 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 921 . . . . . . . . . . . . 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)))
1210, 11bitri 275 . . . . . . . . . . . 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)))
139, 12sylib 218 . . . . . . . . . . 11 ((𝐴 No 𝐵 No ) → (dom 𝐴 = dom 𝐵 ∨ (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴)))
1413ord 865 . . . . . . . . . 10 ((𝐴 No 𝐵 No ) → (¬ dom 𝐴 = dom 𝐵 → (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴)))
15 noseponlem 27709 . . . . . . . . . . . 12 ((𝐴 No 𝐵 No ∧ dom 𝐴 ∈ dom 𝐵) → ¬ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥))
16153expia 1122 . . . . . . . . . . 11 ((𝐴 No 𝐵 No ) → (dom 𝐴 ∈ dom 𝐵 → ¬ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)))
17 noseponlem 27709 . . . . . . . . . . . . . 14 ((𝐵 No 𝐴 No ∧ dom 𝐵 ∈ dom 𝐴) → ¬ ∀𝑥 ∈ On (𝐵𝑥) = (𝐴𝑥))
18 eqcom 2744 . . . . . . . . . . . . . . 15 ((𝐴𝑥) = (𝐵𝑥) ↔ (𝐵𝑥) = (𝐴𝑥))
1918ralbii 3093 . . . . . . . . . . . . . 14 (∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥) ↔ ∀𝑥 ∈ On (𝐵𝑥) = (𝐴𝑥))
2017, 19sylnibr 329 . . . . . . . . . . . . 13 ((𝐵 No 𝐴 No ∧ dom 𝐵 ∈ dom 𝐴) → ¬ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥))
21203expia 1122 . . . . . . . . . . . 12 ((𝐵 No 𝐴 No ) → (dom 𝐵 ∈ dom 𝐴 → ¬ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)))
2221ancoms 458 . . . . . . . . . . 11 ((𝐴 No 𝐵 No ) → (dom 𝐵 ∈ dom 𝐴 → ¬ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)))
2316, 22jaod 860 . . . . . . . . . 10 ((𝐴 No 𝐵 No ) → ((dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴) → ¬ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)))
2414, 23syld 47 . . . . . . . . 9 ((𝐴 No 𝐵 No ) → (¬ dom 𝐴 = dom 𝐵 → ¬ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)))
2524con4d 115 . . . . . . . 8 ((𝐴 No 𝐵 No ) → (∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥) → dom 𝐴 = dom 𝐵))
26253impia 1118 . . . . . . 7 ((𝐴 No 𝐵 No ∧ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)) → dom 𝐴 = dom 𝐵)
27 ordsson 7803 . . . . . . . . . 10 (Ord dom 𝐴 → dom 𝐴 ⊆ On)
28 ssralv 4052 . . . . . . . . . 10 (dom 𝐴 ⊆ On → (∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥) → ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥)))
296, 27, 283syl 18 . . . . . . . . 9 (𝐴 No → (∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥) → ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥)))
3029adantr 480 . . . . . . . 8 ((𝐴 No 𝐵 No ) → (∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥) → ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥)))
31303impia 1118 . . . . . . 7 ((𝐴 No 𝐵 No ∧ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)) → ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))
32 nofun 27694 . . . . . . . . 9 (𝐴 No → Fun 𝐴)
33323ad2ant1 1134 . . . . . . . 8 ((𝐴 No 𝐵 No ∧ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)) → Fun 𝐴)
34 nofun 27694 . . . . . . . . 9 (𝐵 No → Fun 𝐵)
35343ad2ant2 1135 . . . . . . . 8 ((𝐴 No 𝐵 No ∧ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)) → Fun 𝐵)
36 eqfunfv 7056 . . . . . . . 8 ((Fun 𝐴 ∧ Fun 𝐵) → (𝐴 = 𝐵 ↔ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))))
3733, 35, 36syl2anc 584 . . . . . . 7 ((𝐴 No 𝐵 No ∧ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)) → (𝐴 = 𝐵 ↔ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))))
3826, 31, 37mpbir2and 713 . . . . . 6 ((𝐴 No 𝐵 No ∧ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)) → 𝐴 = 𝐵)
39383expia 1122 . . . . 5 ((𝐴 No 𝐵 No ) → (∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥) → 𝐴 = 𝐵))
405, 39biimtrid 242 . . . 4 ((𝐴 No 𝐵 No ) → (¬ ∃𝑥 ∈ On (𝐴𝑥) ≠ (𝐵𝑥) → 𝐴 = 𝐵))
4140necon1ad 2957 . . 3 ((𝐴 No 𝐵 No ) → (𝐴𝐵 → ∃𝑥 ∈ On (𝐴𝑥) ≠ (𝐵𝑥)))
42413impia 1118 . 2 ((𝐴 No 𝐵 No 𝐴𝐵) → ∃𝑥 ∈ On (𝐴𝑥) ≠ (𝐵𝑥))
43 onintrab2 7817 . 2 (∃𝑥 ∈ On (𝐴𝑥) ≠ (𝐵𝑥) ↔ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ On)
4442, 43sylib 218 1 ((𝐴 No 𝐵 No 𝐴𝐵) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ On)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848  w3o 1086  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  {crab 3436  wss 3951   cint 4946  dom cdm 5685  Ord word 6383  Oncon0 6384  Fun wfun 6555  cfv 6561   No csur 27684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-1o 8506  df-2o 8507  df-no 27687
This theorem is referenced by:  nosepeq  27730  nosepssdm  27731  nodenselem4  27732  noresle  27742  nosupbnd2lem1  27760  noinfbnd2lem1  27775  noetasuplem4  27781  noetainflem4  27785
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