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Theorem nosepon 27036
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 27024 . . . . . . . . . . . . 13 (𝐴 No → Ord dom 𝐴)
7 nodmord 27024 . . . . . . . . . . . . 13 (𝐵 No → Ord dom 𝐵)
8 ordtri3or 6353 . . . . . . . . . . . . 13 ((Ord dom 𝐴 ∧ Ord dom 𝐵) → (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐴 = dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴))
96, 7, 8syl2an 597 . . . . . . . . . . . 12 ((𝐴 No 𝐵 No ) → (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐴 = dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴))
10 3orass 1091 . . . . . . . . . . . . 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 920 . . . . . . . . . . . . 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 217 . . . . . . . . . . 11 ((𝐴 No 𝐵 No ) → (dom 𝐴 = dom 𝐵 ∨ (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴)))
1413ord 863 . . . . . . . . . 10 ((𝐴 No 𝐵 No ) → (¬ dom 𝐴 = dom 𝐵 → (dom 𝐴 ∈ dom 𝐵 ∨ dom 𝐵 ∈ dom 𝐴)))
15 noseponlem 27035 . . . . . . . . . . . 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 27035 . . . . . . . . . . . . . 14 ((𝐵 No 𝐴 No ∧ dom 𝐵 ∈ dom 𝐴) → ¬ ∀𝑥 ∈ On (𝐵𝑥) = (𝐴𝑥))
18 eqcom 2740 . . . . . . . . . . . . . . 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 460 . . . . . . . . . . 11 ((𝐴 No 𝐵 No ) → (dom 𝐵 ∈ dom 𝐴 → ¬ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)))
2316, 22jaod 858 . . . . . . . . . 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 7721 . . . . . . . . . 10 (Ord dom 𝐴 → dom 𝐴 ⊆ On)
28 ssralv 4014 . . . . . . . . . 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 482 . . . . . . . 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 27020 . . . . . . . . 9 (𝐴 No → Fun 𝐴)
33323ad2ant1 1134 . . . . . . . 8 ((𝐴 No 𝐵 No ∧ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)) → Fun 𝐴)
34 nofun 27020 . . . . . . . . 9 (𝐵 No → Fun 𝐵)
35343ad2ant2 1135 . . . . . . . 8 ((𝐴 No 𝐵 No ∧ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)) → Fun 𝐵)
36 eqfunfv 6991 . . . . . . . 8 ((Fun 𝐴 ∧ Fun 𝐵) → (𝐴 = 𝐵 ↔ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))))
3733, 35, 36syl2anc 585 . . . . . . 7 ((𝐴 No 𝐵 No ∧ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥)) → (𝐴 = 𝐵 ↔ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))))
3826, 31, 37mpbir2and 712 . . . . . 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 241 . . . 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 7736 . 2 (∃𝑥 ∈ On (𝐴𝑥) ≠ (𝐵𝑥) ↔ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ On)
4442, 43sylib 217 1 ((𝐴 No 𝐵 No 𝐴𝐵) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ On)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846  w3o 1087  w3a 1088   = wceq 1542  wcel 2107  wne 2940  wral 3061  wrex 3070  {crab 3406  wss 3914   cint 4911  dom cdm 5637  Ord word 6320  Oncon0 6321  Fun wfun 6494  cfv 6500   No csur 27011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-1o 8416  df-2o 8417  df-no 27014
This theorem is referenced by:  nosepeq  27056  nosepssdm  27057  nodenselem4  27058  noresle  27068  nosupbnd2lem1  27086  noinfbnd2lem1  27101  noetasuplem4  27107  noetainflem4  27111
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