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Theorem nosepon 32154
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 2944 . . . . . . . 8 ((𝐴𝑥) ≠ (𝐵𝑥) ↔ ¬ (𝐴𝑥) = (𝐵𝑥))
21rexbii 3189 . . . . . . 7 (∃𝑥 ∈ On (𝐴𝑥) ≠ (𝐵𝑥) ↔ ∃𝑥 ∈ On ¬ (𝐴𝑥) = (𝐵𝑥))
32notbii 309 . . . . . 6 (¬ ∃𝑥 ∈ On (𝐴𝑥) ≠ (𝐵𝑥) ↔ ¬ ∃𝑥 ∈ On ¬ (𝐴𝑥) = (𝐵𝑥))
4 dfral2 3142 . . . . . 6 (∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥) ↔ ¬ ∃𝑥 ∈ On ¬ (𝐴𝑥) = (𝐵𝑥))
53, 4bitr4i 267 . . . . 5 (¬ ∃𝑥 ∈ On (𝐴𝑥) ≠ (𝐵𝑥) ↔ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥))
6 nodmord 32142 . . . . . . . . . . . . 13 (𝐴 No → Ord dom 𝐴)
7 nodmord 32142 . . . . . . . . . . . . 13 (𝐵 No → Ord dom 𝐵)
8 ordtri3or 5897 . . . . . . . . . . . . 13 ((Ord dom 𝐴 ∧ Ord dom 𝐵) → (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐴 = dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴))
96, 7, 8syl2an 583 . . . . . . . . . . . 12 ((𝐴 No 𝐵 No ) → (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐴 = dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴))
10 3orass 1074 . . . . . . . . . . . . 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 906 . . . . . . . . . . . . 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 853 . . . . . . . . . 10 ((𝐴 No 𝐵 No ) → (¬ dom 𝐴 = dom 𝐵 → (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴)))
15 noseponlem 32153 . . . . . . . . . . . 12 ((𝐴 No 𝐵 No ∧ dom 𝐴 ∈ dom 𝐵) → ¬ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥))
16153expia 1114 . . . . . . . . . . 11 ((𝐴 No 𝐵 No ) → (dom 𝐴 ∈ dom 𝐵 → ¬ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)))
17 noseponlem 32153 . . . . . . . . . . . . . 14 ((𝐵 No 𝐴 No ∧ dom 𝐵 ∈ dom 𝐴) → ¬ ∀𝑥 ∈ On (𝐵𝑥) = (𝐴𝑥))
18 eqcom 2778 . . . . . . . . . . . . . . 15 ((𝐴𝑥) = (𝐵𝑥) ↔ (𝐵𝑥) = (𝐴𝑥))
1918ralbii 3129 . . . . . . . . . . . . . 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 1114 . . . . . . . . . . . 12 ((𝐵 No 𝐴 No ) → (dom 𝐵 ∈ dom 𝐴 → ¬ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)))
2221ancoms 446 . . . . . . . . . . 11 ((𝐴 No 𝐵 No ) → (dom 𝐵 ∈ dom 𝐴 → ¬ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)))
2316, 22jaod 848 . . . . . . . . . 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 1109 . . . . . . 7 ((𝐴 No 𝐵 No ∧ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)) → dom 𝐴 = dom 𝐵)
27 ordsson 7139 . . . . . . . . . 10 (Ord dom 𝐴 → dom 𝐴 ⊆ On)
28 ssralv 3815 . . . . . . . . . 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 466 . . . . . . . 8 ((𝐴 No 𝐵 No ) → (∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥) → ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥)))
31303impia 1109 . . . . . . 7 ((𝐴 No 𝐵 No ∧ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)) → ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))
32 nofun 32138 . . . . . . . . 9 (𝐴 No → Fun 𝐴)
33323ad2ant1 1127 . . . . . . . 8 ((𝐴 No 𝐵 No ∧ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)) → Fun 𝐴)
34 nofun 32138 . . . . . . . . 9 (𝐵 No → Fun 𝐵)
35343ad2ant2 1128 . . . . . . . 8 ((𝐴 No 𝐵 No ∧ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)) → Fun 𝐵)
36 eqfunfv 6461 . . . . . . . 8 ((Fun 𝐴 ∧ Fun 𝐵) → (𝐴 = 𝐵 ↔ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))))
3733, 35, 36syl2anc 573 . . . . . . 7 ((𝐴 No 𝐵 No ∧ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)) → (𝐴 = 𝐵 ↔ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))))
3826, 31, 37mpbir2and 692 . . . . . 6 ((𝐴 No 𝐵 No ∧ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)) → 𝐴 = 𝐵)
39383expia 1114 . . . . 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 2960 . . 3 ((𝐴 No 𝐵 No ) → (𝐴𝐵 → ∃𝑥 ∈ On (𝐴𝑥) ≠ (𝐵𝑥)))
42413impia 1109 . 2 ((𝐴 No 𝐵 No 𝐴𝐵) → ∃𝑥 ∈ On (𝐴𝑥) ≠ (𝐵𝑥))
43 onintrab2 7152 . 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  wa 382  wo 836  w3o 1070  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wral 3061  wrex 3062  {crab 3065  wss 3723   cint 4612  dom cdm 5250  Ord word 5864  Oncon0 5865  Fun wfun 6024  cfv 6030   No csur 32129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7099
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-ord 5868  df-on 5869  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-1o 7716  df-2o 7717  df-no 32132
This theorem is referenced by:  nosepeq  32171  nosepssdm  32172  nodenselem4  32173  noresle  32182  nosupbnd2lem1  32197  noetalem3  32201
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