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Theorem nnneo 8272
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 7580 . . . 4 (𝐴 ∈ ω → 𝐴 ∈ On)
2 onnbtwn 6277 . . . 4 (𝐴 ∈ On → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))
31, 2syl 17 . . 3 (𝐴 ∈ ω → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))
433ad2ant1 1129 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2o ·o 𝐴)) → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))
5 suceq 6251 . . . . 5 (𝐶 = (2o ·o 𝐴) → suc 𝐶 = suc (2o ·o 𝐴))
65eqeq1d 2823 . . . 4 (𝐶 = (2o ·o 𝐴) → (suc 𝐶 = (2o ·o 𝐵) ↔ suc (2o ·o 𝐴) = (2o ·o 𝐵)))
763ad2ant3 1131 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2o ·o 𝐴)) → (suc 𝐶 = (2o ·o 𝐵) ↔ suc (2o ·o 𝐴) = (2o ·o 𝐵)))
8 ovex 7183 . . . . . . . 8 (2o ·o 𝐴) ∈ V
98sucid 6265 . . . . . . 7 (2o ·o 𝐴) ∈ suc (2o ·o 𝐴)
10 eleq2 2901 . . . . . . 7 (suc (2o ·o 𝐴) = (2o ·o 𝐵) → ((2o ·o 𝐴) ∈ suc (2o ·o 𝐴) ↔ (2o ·o 𝐴) ∈ (2o ·o 𝐵)))
119, 10mpbii 235 . . . . . 6 (suc (2o ·o 𝐴) = (2o ·o 𝐵) → (2o ·o 𝐴) ∈ (2o ·o 𝐵))
12 2onn 8260 . . . . . . . 8 2o ∈ ω
13 nnmord 8252 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 2o ∈ ω) → ((𝐴𝐵 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐴) ∈ (2o ·o 𝐵)))
1412, 13mp3an3 1446 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴𝐵 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐴) ∈ (2o ·o 𝐵)))
15 simpl 485 . . . . . . 7 ((𝐴𝐵 ∧ ∅ ∈ 2o) → 𝐴𝐵)
1614, 15syl6bir 256 . . . . . 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 487 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → suc (2o ·o 𝐴) = (2o ·o 𝐵))
19 nnmcl 8232 . . . . . . . . . . . . 13 ((2o ∈ ω ∧ 𝐴 ∈ ω) → (2o ·o 𝐴) ∈ ω)
2012, 19mpan 688 . . . . . . . . . . . 12 (𝐴 ∈ ω → (2o ·o 𝐴) ∈ ω)
21 nnon 7580 . . . . . . . . . . . 12 ((2o ·o 𝐴) ∈ ω → (2o ·o 𝐴) ∈ On)
22 oa1suc 8150 . . . . . . . . . . . 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 8104 . . . . . . . . . . . . . . 15 1o ∈ V
2524sucid 6265 . . . . . . . . . . . . . 14 1o ∈ suc 1o
26 df-2o 8097 . . . . . . . . . . . . . 14 2o = suc 1o
2725, 26eleqtrri 2912 . . . . . . . . . . . . 13 1o ∈ 2o
28 1onn 8259 . . . . . . . . . . . . . 14 1o ∈ ω
29 nnaord 8239 . . . . . . . . . . . . . 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 1461 . . . . . . . . . . . . 13 (𝐴 ∈ ω → (1o ∈ 2o ↔ ((2o ·o 𝐴) +o 1o) ∈ ((2o ·o 𝐴) +o 2o)))
3127, 30mpbii 235 . . . . . . . . . . . 12 (𝐴 ∈ ω → ((2o ·o 𝐴) +o 1o) ∈ ((2o ·o 𝐴) +o 2o))
32 nnmsuc 8227 . . . . . . . . . . . . 13 ((2o ∈ ω ∧ 𝐴 ∈ ω) → (2o ·o suc 𝐴) = ((2o ·o 𝐴) +o 2o))
3312, 32mpan 688 . . . . . . . . . . . 12 (𝐴 ∈ ω → (2o ·o suc 𝐴) = ((2o ·o 𝐴) +o 2o))
3431, 33eleqtrrd 2916 . . . . . . . . . . 11 (𝐴 ∈ ω → ((2o ·o 𝐴) +o 1o) ∈ (2o ·o suc 𝐴))
3523, 34eqeltrrd 2914 . . . . . . . . . 10 (𝐴 ∈ ω → suc (2o ·o 𝐴) ∈ (2o ·o suc 𝐴))
3635ad2antrr 724 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → suc (2o ·o 𝐴) ∈ (2o ·o suc 𝐴))
3718, 36eqeltrrd 2914 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → (2o ·o 𝐵) ∈ (2o ·o suc 𝐴))
38 peano2 7596 . . . . . . . . . . 11 (𝐴 ∈ ω → suc 𝐴 ∈ ω)
39 nnmord 8252 . . . . . . . . . . . 12 ((𝐵 ∈ ω ∧ suc 𝐴 ∈ ω ∧ 2o ∈ ω) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
4012, 39mp3an3 1446 . . . . . . . . . . 11 ((𝐵 ∈ ω ∧ suc 𝐴 ∈ ω) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
4138, 40sylan2 594 . . . . . . . . . 10 ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
4241ancoms 461 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
4342adantr 483 . . . . . . . 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 259 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → (𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o))
4544simpld 497 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → 𝐵 ∈ suc 𝐴)
4645ex 415 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc (2o ·o 𝐴) = (2o ·o 𝐵) → 𝐵 ∈ suc 𝐴))
4717, 46jcad 515 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc (2o ·o 𝐴) = (2o ·o 𝐵) → (𝐴𝐵𝐵 ∈ suc 𝐴)))
48473adant3 1128 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2o ·o 𝐴)) → (suc (2o ·o 𝐴) = (2o ·o 𝐵) → (𝐴𝐵𝐵 ∈ suc 𝐴)))
497, 48sylbid 242 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2o ·o 𝐴)) → (suc 𝐶 = (2o ·o 𝐵) → (𝐴𝐵𝐵 ∈ suc 𝐴)))
504, 49mtod 200 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 208  wa 398  w3a 1083   = wceq 1533  wcel 2110  c0 4291  Oncon0 6186  suc csuc 6188  (class class class)co 7150  ωcom 7574  1oc1o 8089  2oc2o 8090   +o coa 8093   ·o comu 8094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-omul 8101
This theorem is referenced by:  nneob  8273
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