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Theorem nnneo 8267
Description: If a natural number is even, its successor is odd. (Contributed by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
nnneo ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2o ·o 𝐴)) → ¬ suc 𝐶 = (2o ·o 𝐵))

Proof of Theorem nnneo
StepHypRef Expression
1 nnon 7575 . . . 4 (𝐴 ∈ ω → 𝐴 ∈ On)
2 onnbtwn 6275 . . . 4 (𝐴 ∈ On → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))
31, 2syl 17 . . 3 (𝐴 ∈ ω → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))
433ad2ant1 1125 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2o ·o 𝐴)) → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))
5 suceq 6249 . . . . 5 (𝐶 = (2o ·o 𝐴) → suc 𝐶 = suc (2o ·o 𝐴))
65eqeq1d 2820 . . . 4 (𝐶 = (2o ·o 𝐴) → (suc 𝐶 = (2o ·o 𝐵) ↔ suc (2o ·o 𝐴) = (2o ·o 𝐵)))
763ad2ant3 1127 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2o ·o 𝐴)) → (suc 𝐶 = (2o ·o 𝐵) ↔ suc (2o ·o 𝐴) = (2o ·o 𝐵)))
8 ovex 7178 . . . . . . . 8 (2o ·o 𝐴) ∈ V
98sucid 6263 . . . . . . 7 (2o ·o 𝐴) ∈ suc (2o ·o 𝐴)
10 eleq2 2898 . . . . . . 7 (suc (2o ·o 𝐴) = (2o ·o 𝐵) → ((2o ·o 𝐴) ∈ suc (2o ·o 𝐴) ↔ (2o ·o 𝐴) ∈ (2o ·o 𝐵)))
119, 10mpbii 234 . . . . . 6 (suc (2o ·o 𝐴) = (2o ·o 𝐵) → (2o ·o 𝐴) ∈ (2o ·o 𝐵))
12 2onn 8255 . . . . . . . 8 2o ∈ ω
13 nnmord 8247 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 2o ∈ ω) → ((𝐴𝐵 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐴) ∈ (2o ·o 𝐵)))
1412, 13mp3an3 1441 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴𝐵 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐴) ∈ (2o ·o 𝐵)))
15 simpl 483 . . . . . . 7 ((𝐴𝐵 ∧ ∅ ∈ 2o) → 𝐴𝐵)
1614, 15syl6bir 255 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((2o ·o 𝐴) ∈ (2o ·o 𝐵) → 𝐴𝐵))
1711, 16syl5 34 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc (2o ·o 𝐴) = (2o ·o 𝐵) → 𝐴𝐵))
18 simpr 485 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → suc (2o ·o 𝐴) = (2o ·o 𝐵))
19 nnmcl 8227 . . . . . . . . . . . . 13 ((2o ∈ ω ∧ 𝐴 ∈ ω) → (2o ·o 𝐴) ∈ ω)
2012, 19mpan 686 . . . . . . . . . . . 12 (𝐴 ∈ ω → (2o ·o 𝐴) ∈ ω)
21 nnon 7575 . . . . . . . . . . . 12 ((2o ·o 𝐴) ∈ ω → (2o ·o 𝐴) ∈ On)
22 oa1suc 8145 . . . . . . . . . . . 12 ((2o ·o 𝐴) ∈ On → ((2o ·o 𝐴) +o 1o) = suc (2o ·o 𝐴))
2320, 21, 223syl 18 . . . . . . . . . . 11 (𝐴 ∈ ω → ((2o ·o 𝐴) +o 1o) = suc (2o ·o 𝐴))
24 1oex 8099 . . . . . . . . . . . . . . 15 1o ∈ V
2524sucid 6263 . . . . . . . . . . . . . 14 1o ∈ suc 1o
26 df-2o 8092 . . . . . . . . . . . . . 14 2o = suc 1o
2725, 26eleqtrri 2909 . . . . . . . . . . . . 13 1o ∈ 2o
28 1onn 8254 . . . . . . . . . . . . . 14 1o ∈ ω
29 nnaord 8234 . . . . . . . . . . . . . 14 ((1o ∈ ω ∧ 2o ∈ ω ∧ (2o ·o 𝐴) ∈ ω) → (1o ∈ 2o ↔ ((2o ·o 𝐴) +o 1o) ∈ ((2o ·o 𝐴) +o 2o)))
3028, 12, 20, 29mp3an12i 1456 . . . . . . . . . . . . 13 (𝐴 ∈ ω → (1o ∈ 2o ↔ ((2o ·o 𝐴) +o 1o) ∈ ((2o ·o 𝐴) +o 2o)))
3127, 30mpbii 234 . . . . . . . . . . . 12 (𝐴 ∈ ω → ((2o ·o 𝐴) +o 1o) ∈ ((2o ·o 𝐴) +o 2o))
32 nnmsuc 8222 . . . . . . . . . . . . 13 ((2o ∈ ω ∧ 𝐴 ∈ ω) → (2o ·o suc 𝐴) = ((2o ·o 𝐴) +o 2o))
3312, 32mpan 686 . . . . . . . . . . . 12 (𝐴 ∈ ω → (2o ·o suc 𝐴) = ((2o ·o 𝐴) +o 2o))
3431, 33eleqtrrd 2913 . . . . . . . . . . 11 (𝐴 ∈ ω → ((2o ·o 𝐴) +o 1o) ∈ (2o ·o suc 𝐴))
3523, 34eqeltrrd 2911 . . . . . . . . . 10 (𝐴 ∈ ω → suc (2o ·o 𝐴) ∈ (2o ·o suc 𝐴))
3635ad2antrr 722 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → suc (2o ·o 𝐴) ∈ (2o ·o suc 𝐴))
3718, 36eqeltrrd 2911 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → (2o ·o 𝐵) ∈ (2o ·o suc 𝐴))
38 peano2 7591 . . . . . . . . . . 11 (𝐴 ∈ ω → suc 𝐴 ∈ ω)
39 nnmord 8247 . . . . . . . . . . . 12 ((𝐵 ∈ ω ∧ suc 𝐴 ∈ ω ∧ 2o ∈ ω) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
4012, 39mp3an3 1441 . . . . . . . . . . 11 ((𝐵 ∈ ω ∧ suc 𝐴 ∈ ω) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
4138, 40sylan2 592 . . . . . . . . . 10 ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
4241ancoms 459 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
4342adantr 481 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
4437, 43mpbird 258 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → (𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o))
4544simpld 495 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → 𝐵 ∈ suc 𝐴)
4645ex 413 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc (2o ·o 𝐴) = (2o ·o 𝐵) → 𝐵 ∈ suc 𝐴))
4717, 46jcad 513 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc (2o ·o 𝐴) = (2o ·o 𝐵) → (𝐴𝐵𝐵 ∈ suc 𝐴)))
48473adant3 1124 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2o ·o 𝐴)) → (suc (2o ·o 𝐴) = (2o ·o 𝐵) → (𝐴𝐵𝐵 ∈ suc 𝐴)))
497, 48sylbid 241 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2o ·o 𝐴)) → (suc 𝐶 = (2o ·o 𝐵) → (𝐴𝐵𝐵 ∈ suc 𝐴)))
504, 49mtod 199 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2o ·o 𝐴)) → ¬ suc 𝐶 = (2o ·o 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  c0 4288  Oncon0 6184  suc csuc 6186  (class class class)co 7145  ωcom 7569  1oc1o 8084  2oc2o 8085   +o coa 8088   ·o comu 8089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-omul 8096
This theorem is referenced by:  nneob  8268
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