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Theorem nosepon 31943
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 2824 . . . . . . . 8 ((𝐴𝑥) ≠ (𝐵𝑥) ↔ ¬ (𝐴𝑥) = (𝐵𝑥))
21rexbii 3070 . . . . . . 7 (∃𝑥 ∈ On (𝐴𝑥) ≠ (𝐵𝑥) ↔ ∃𝑥 ∈ On ¬ (𝐴𝑥) = (𝐵𝑥))
32notbii 309 . . . . . 6 (¬ ∃𝑥 ∈ On (𝐴𝑥) ≠ (𝐵𝑥) ↔ ¬ ∃𝑥 ∈ On ¬ (𝐴𝑥) = (𝐵𝑥))
4 dfral2 3023 . . . . . 6 (∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥) ↔ ¬ ∃𝑥 ∈ On ¬ (𝐴𝑥) = (𝐵𝑥))
53, 4bitr4i 267 . . . . 5 (¬ ∃𝑥 ∈ On (𝐴𝑥) ≠ (𝐵𝑥) ↔ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥))
6 nodmord 31931 . . . . . . . . . . . . 13 (𝐴 No → Ord dom 𝐴)
7 nodmord 31931 . . . . . . . . . . . . 13 (𝐵 No → Ord dom 𝐵)
8 ordtri3or 5793 . . . . . . . . . . . . 13 ((Ord dom 𝐴 ∧ Ord dom 𝐵) → (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐴 = dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴))
96, 7, 8syl2an 493 . . . . . . . . . . . 12 ((𝐴 No 𝐵 No ) → (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐴 = dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴))
10 3orass 1057 . . . . . . . . . . . . 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 544 . . . . . . . . . . . . 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 264 . . . . . . . . . . . 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 208 . . . . . . . . . . 11 ((𝐴 No 𝐵 No ) → (dom 𝐴 = dom 𝐵 ∨ (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴)))
1413ord 391 . . . . . . . . . 10 ((𝐴 No 𝐵 No ) → (¬ dom 𝐴 = dom 𝐵 → (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴)))
15 noseponlem 31942 . . . . . . . . . . . 12 ((𝐴 No 𝐵 No ∧ dom 𝐴 ∈ dom 𝐵) → ¬ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥))
16153expia 1286 . . . . . . . . . . 11 ((𝐴 No 𝐵 No ) → (dom 𝐴 ∈ dom 𝐵 → ¬ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)))
17 noseponlem 31942 . . . . . . . . . . . . . 14 ((𝐵 No 𝐴 No ∧ dom 𝐵 ∈ dom 𝐴) → ¬ ∀𝑥 ∈ On (𝐵𝑥) = (𝐴𝑥))
18 eqcom 2658 . . . . . . . . . . . . . . 15 ((𝐴𝑥) = (𝐵𝑥) ↔ (𝐵𝑥) = (𝐴𝑥))
1918ralbii 3009 . . . . . . . . . . . . . 14 (∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥) ↔ ∀𝑥 ∈ On (𝐵𝑥) = (𝐴𝑥))
2017, 19sylnibr 318 . . . . . . . . . . . . 13 ((𝐵 No 𝐴 No ∧ dom 𝐵 ∈ dom 𝐴) → ¬ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥))
21203expia 1286 . . . . . . . . . . . 12 ((𝐵 No 𝐴 No ) → (dom 𝐵 ∈ dom 𝐴 → ¬ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)))
2221ancoms 468 . . . . . . . . . . 11 ((𝐴 No 𝐵 No ) → (dom 𝐵 ∈ dom 𝐴 → ¬ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)))
2316, 22jaod 394 . . . . . . . . . 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 114 . . . . . . . 8 ((𝐴 No 𝐵 No ) → (∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥) → dom 𝐴 = dom 𝐵))
26253impia 1280 . . . . . . 7 ((𝐴 No 𝐵 No ∧ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)) → dom 𝐴 = dom 𝐵)
27 ordsson 7031 . . . . . . . . . 10 (Ord dom 𝐴 → dom 𝐴 ⊆ On)
28 ssralv 3699 . . . . . . . . . 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 1280 . . . . . . 7 ((𝐴 No 𝐵 No ∧ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)) → ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))
32 nofun 31927 . . . . . . . . 9 (𝐴 No → Fun 𝐴)
33323ad2ant1 1102 . . . . . . . 8 ((𝐴 No 𝐵 No ∧ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)) → Fun 𝐴)
34 nofun 31927 . . . . . . . . 9 (𝐵 No → Fun 𝐵)
35343ad2ant2 1103 . . . . . . . 8 ((𝐴 No 𝐵 No ∧ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)) → Fun 𝐵)
36 eqfunfv 6356 . . . . . . . 8 ((Fun 𝐴 ∧ Fun 𝐵) → (𝐴 = 𝐵 ↔ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))))
3733, 35, 36syl2anc 694 . . . . . . 7 ((𝐴 No 𝐵 No ∧ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)) → (𝐴 = 𝐵 ↔ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))))
3826, 31, 37mpbir2and 977 . . . . . 6 ((𝐴 No 𝐵 No ∧ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)) → 𝐴 = 𝐵)
39383expia 1286 . . . . 5 ((𝐴 No 𝐵 No ) → (∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥) → 𝐴 = 𝐵))
405, 39syl5bi 232 . . . 4 ((𝐴 No 𝐵 No ) → (¬ ∃𝑥 ∈ On (𝐴𝑥) ≠ (𝐵𝑥) → 𝐴 = 𝐵))
4140necon1ad 2840 . . 3 ((𝐴 No 𝐵 No ) → (𝐴𝐵 → ∃𝑥 ∈ On (𝐴𝑥) ≠ (𝐵𝑥)))
42413impia 1280 . 2 ((𝐴 No 𝐵 No 𝐴𝐵) → ∃𝑥 ∈ On (𝐴𝑥) ≠ (𝐵𝑥))
43 onintrab2 7044 . 2 (∃𝑥 ∈ On (𝐴𝑥) ≠ (𝐵𝑥) ↔ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ On)
4442, 43sylib 208 1 ((𝐴 No 𝐵 No 𝐴𝐵) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ On)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3o 1053  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  {crab 2945  wss 3607   cint 4507  dom cdm 5143  Ord word 5760  Oncon0 5761  Fun wfun 5920  cfv 5926   No csur 31918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-1o 7605  df-2o 7606  df-no 31921
This theorem is referenced by:  nosepeq  31960  nosepssdm  31961  nodenselem4  31962  noresle  31971  nosupbnd2lem1  31986  noetalem3  31990
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