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Theorem nosepon 31550
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 2791 . . . . . . . 8 ((𝐴𝑥) ≠ (𝐵𝑥) ↔ ¬ (𝐴𝑥) = (𝐵𝑥))
21rexbii 3035 . . . . . . 7 (∃𝑥 ∈ On (𝐴𝑥) ≠ (𝐵𝑥) ↔ ∃𝑥 ∈ On ¬ (𝐴𝑥) = (𝐵𝑥))
32notbii 310 . . . . . 6 (¬ ∃𝑥 ∈ On (𝐴𝑥) ≠ (𝐵𝑥) ↔ ¬ ∃𝑥 ∈ On ¬ (𝐴𝑥) = (𝐵𝑥))
4 dfral2 2989 . . . . . 6 (∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥) ↔ ¬ ∃𝑥 ∈ On ¬ (𝐴𝑥) = (𝐵𝑥))
53, 4bitr4i 267 . . . . 5 (¬ ∃𝑥 ∈ On (𝐴𝑥) ≠ (𝐵𝑥) ↔ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥))
6 nodmord 31534 . . . . . . . . . . . . 13 (𝐴 No → Ord dom 𝐴)
7 nodmord 31534 . . . . . . . . . . . . 13 (𝐵 No → Ord dom 𝐵)
8 ordtri3or 5719 . . . . . . . . . . . . 13 ((Ord dom 𝐴 ∧ Ord dom 𝐵) → (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐴 = dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴))
96, 7, 8syl2an 494 . . . . . . . . . . . 12 ((𝐴 No 𝐵 No ) → (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐴 = dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴))
10 3orass 1039 . . . . . . . . . . . . 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 545 . . . . . . . . . . . . 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 392 . . . . . . . . . 10 ((𝐴 No 𝐵 No ) → (¬ dom 𝐴 = dom 𝐵 → (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴)))
15 noseponlem 31549 . . . . . . . . . . . 12 ((𝐴 No 𝐵 No ∧ dom 𝐴 ∈ dom 𝐵) → ¬ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥))
16153expia 1264 . . . . . . . . . . 11 ((𝐴 No 𝐵 No ) → (dom 𝐴 ∈ dom 𝐵 → ¬ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)))
17 noseponlem 31549 . . . . . . . . . . . . . 14 ((𝐵 No 𝐴 No ∧ dom 𝐵 ∈ dom 𝐴) → ¬ ∀𝑥 ∈ On (𝐵𝑥) = (𝐴𝑥))
18 eqcom 2628 . . . . . . . . . . . . . . 15 ((𝐴𝑥) = (𝐵𝑥) ↔ (𝐵𝑥) = (𝐴𝑥))
1918ralbii 2975 . . . . . . . . . . . . . 14 (∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥) ↔ ∀𝑥 ∈ On (𝐵𝑥) = (𝐴𝑥))
2017, 19sylnibr 319 . . . . . . . . . . . . 13 ((𝐵 No 𝐴 No ∧ dom 𝐵 ∈ dom 𝐴) → ¬ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥))
21203expia 1264 . . . . . . . . . . . 12 ((𝐵 No 𝐴 No ) → (dom 𝐵 ∈ dom 𝐴 → ¬ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)))
2221ancoms 469 . . . . . . . . . . 11 ((𝐴 No 𝐵 No ) → (dom 𝐵 ∈ dom 𝐴 → ¬ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)))
2316, 22jaod 395 . . . . . . . . . 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 1258 . . . . . . 7 ((𝐴 No 𝐵 No ∧ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)) → dom 𝐴 = dom 𝐵)
27 ordsson 6943 . . . . . . . . . 10 (Ord dom 𝐴 → dom 𝐴 ⊆ On)
28 ssralv 3650 . . . . . . . . . 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 481 . . . . . . . 8 ((𝐴 No 𝐵 No ) → (∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥) → ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥)))
31303impia 1258 . . . . . . 7 ((𝐴 No 𝐵 No ∧ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)) → ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))
32 nofun 31530 . . . . . . . . 9 (𝐴 No → Fun 𝐴)
33323ad2ant1 1080 . . . . . . . 8 ((𝐴 No 𝐵 No ∧ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)) → Fun 𝐴)
34 nofun 31530 . . . . . . . . 9 (𝐵 No → Fun 𝐵)
35343ad2ant2 1081 . . . . . . . 8 ((𝐴 No 𝐵 No ∧ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)) → Fun 𝐵)
36 eqfunfv 6277 . . . . . . . 8 ((Fun 𝐴 ∧ Fun 𝐵) → (𝐴 = 𝐵 ↔ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))))
3733, 35, 36syl2anc 692 . . . . . . 7 ((𝐴 No 𝐵 No ∧ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)) → (𝐴 = 𝐵 ↔ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))))
3826, 31, 37mpbir2and 956 . . . . . 6 ((𝐴 No 𝐵 No ∧ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)) → 𝐴 = 𝐵)
39383expia 1264 . . . . 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 2807 . . 3 ((𝐴 No 𝐵 No ) → (𝐴𝐵 → ∃𝑥 ∈ On (𝐴𝑥) ≠ (𝐵𝑥)))
42413impia 1258 . 2 ((𝐴 No 𝐵 No 𝐴𝐵) → ∃𝑥 ∈ On (𝐴𝑥) ≠ (𝐵𝑥))
43 onintrab2 6956 . 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 383  wa 384  w3o 1035  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  wss 3559   cint 4445  dom cdm 5079  Ord word 5686  Oncon0 5687  Fun wfun 5846  cfv 5852   No csur 31521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5690  df-on 5691  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-1o 7512  df-2o 7513  df-no 31524
This theorem is referenced by:  noreslege  31598
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