MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oneo Structured version   Visualization version   GIF version

Theorem oneo 7870
Description: If an ordinal number is even, its successor is odd. (Contributed by NM, 26-Jan-2006.)
Assertion
Ref Expression
oneo ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2𝑜 ·𝑜 𝐴)) → ¬ suc 𝐶 = (2𝑜 ·𝑜 𝐵))

Proof of Theorem oneo
StepHypRef Expression
1 onnbtwn 6001 . . 3 (𝐴 ∈ On → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))
213ad2ant1 1163 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2𝑜 ·𝑜 𝐴)) → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))
3 suceq 5975 . . . . 5 (𝐶 = (2𝑜 ·𝑜 𝐴) → suc 𝐶 = suc (2𝑜 ·𝑜 𝐴))
43eqeq1d 2767 . . . 4 (𝐶 = (2𝑜 ·𝑜 𝐴) → (suc 𝐶 = (2𝑜 ·𝑜 𝐵) ↔ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)))
543ad2ant3 1165 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2𝑜 ·𝑜 𝐴)) → (suc 𝐶 = (2𝑜 ·𝑜 𝐵) ↔ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)))
6 ovex 6878 . . . . . . . 8 (2𝑜 ·𝑜 𝐴) ∈ V
76sucid 5989 . . . . . . 7 (2𝑜 ·𝑜 𝐴) ∈ suc (2𝑜 ·𝑜 𝐴)
8 eleq2 2833 . . . . . . 7 (suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵) → ((2𝑜 ·𝑜 𝐴) ∈ suc (2𝑜 ·𝑜 𝐴) ↔ (2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 𝐵)))
97, 8mpbii 224 . . . . . 6 (suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵) → (2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 𝐵))
10 2on 7777 . . . . . . . 8 2𝑜 ∈ On
11 omord 7857 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 2𝑜 ∈ On) → ((𝐴𝐵 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 𝐵)))
1210, 11mp3an3 1574 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴𝐵 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 𝐵)))
13 simpl 474 . . . . . . 7 ((𝐴𝐵 ∧ ∅ ∈ 2𝑜) → 𝐴𝐵)
1412, 13syl6bir 245 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 𝐵) → 𝐴𝐵))
159, 14syl5 34 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵) → 𝐴𝐵))
16 simpr 477 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)) → suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵))
17 omcl 7825 . . . . . . . . . . . . 13 ((2𝑜 ∈ On ∧ 𝐴 ∈ On) → (2𝑜 ·𝑜 𝐴) ∈ On)
1810, 17mpan 681 . . . . . . . . . . . 12 (𝐴 ∈ On → (2𝑜 ·𝑜 𝐴) ∈ On)
19 oa1suc 7820 . . . . . . . . . . . 12 ((2𝑜 ·𝑜 𝐴) ∈ On → ((2𝑜 ·𝑜 𝐴) +𝑜 1𝑜) = suc (2𝑜 ·𝑜 𝐴))
2018, 19syl 17 . . . . . . . . . . 11 (𝐴 ∈ On → ((2𝑜 ·𝑜 𝐴) +𝑜 1𝑜) = suc (2𝑜 ·𝑜 𝐴))
21 1oex 7776 . . . . . . . . . . . . . . 15 1𝑜 ∈ V
2221sucid 5989 . . . . . . . . . . . . . 14 1𝑜 ∈ suc 1𝑜
23 df-2o 7769 . . . . . . . . . . . . . 14 2𝑜 = suc 1𝑜
2422, 23eleqtrri 2843 . . . . . . . . . . . . 13 1𝑜 ∈ 2𝑜
25 1on 7775 . . . . . . . . . . . . . . 15 1𝑜 ∈ On
26 oaord 7836 . . . . . . . . . . . . . . 15 ((1𝑜 ∈ On ∧ 2𝑜 ∈ On ∧ (2𝑜 ·𝑜 𝐴) ∈ On) → (1𝑜 ∈ 2𝑜 ↔ ((2𝑜 ·𝑜 𝐴) +𝑜 1𝑜) ∈ ((2𝑜 ·𝑜 𝐴) +𝑜 2𝑜)))
2725, 10, 26mp3an12 1575 . . . . . . . . . . . . . 14 ((2𝑜 ·𝑜 𝐴) ∈ On → (1𝑜 ∈ 2𝑜 ↔ ((2𝑜 ·𝑜 𝐴) +𝑜 1𝑜) ∈ ((2𝑜 ·𝑜 𝐴) +𝑜 2𝑜)))
2818, 27syl 17 . . . . . . . . . . . . 13 (𝐴 ∈ On → (1𝑜 ∈ 2𝑜 ↔ ((2𝑜 ·𝑜 𝐴) +𝑜 1𝑜) ∈ ((2𝑜 ·𝑜 𝐴) +𝑜 2𝑜)))
2924, 28mpbii 224 . . . . . . . . . . . 12 (𝐴 ∈ On → ((2𝑜 ·𝑜 𝐴) +𝑜 1𝑜) ∈ ((2𝑜 ·𝑜 𝐴) +𝑜 2𝑜))
30 omsuc 7815 . . . . . . . . . . . . 13 ((2𝑜 ∈ On ∧ 𝐴 ∈ On) → (2𝑜 ·𝑜 suc 𝐴) = ((2𝑜 ·𝑜 𝐴) +𝑜 2𝑜))
3110, 30mpan 681 . . . . . . . . . . . 12 (𝐴 ∈ On → (2𝑜 ·𝑜 suc 𝐴) = ((2𝑜 ·𝑜 𝐴) +𝑜 2𝑜))
3229, 31eleqtrrd 2847 . . . . . . . . . . 11 (𝐴 ∈ On → ((2𝑜 ·𝑜 𝐴) +𝑜 1𝑜) ∈ (2𝑜 ·𝑜 suc 𝐴))
3320, 32eqeltrrd 2845 . . . . . . . . . 10 (𝐴 ∈ On → suc (2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 suc 𝐴))
3433ad2antrr 717 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)) → suc (2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 suc 𝐴))
3516, 34eqeltrrd 2845 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)) → (2𝑜 ·𝑜 𝐵) ∈ (2𝑜 ·𝑜 suc 𝐴))
36 suceloni 7215 . . . . . . . . . . 11 (𝐴 ∈ On → suc 𝐴 ∈ On)
37 omord 7857 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ suc 𝐴 ∈ On ∧ 2𝑜 ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐵) ∈ (2𝑜 ·𝑜 suc 𝐴)))
3810, 37mp3an3 1574 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ suc 𝐴 ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐵) ∈ (2𝑜 ·𝑜 suc 𝐴)))
3936, 38sylan2 586 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐵) ∈ (2𝑜 ·𝑜 suc 𝐴)))
4039ancoms 450 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐵) ∈ (2𝑜 ·𝑜 suc 𝐴)))
4140adantr 472 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐵) ∈ (2𝑜 ·𝑜 suc 𝐴)))
4235, 41mpbird 248 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)) → (𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2𝑜))
4342simpld 488 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)) → 𝐵 ∈ suc 𝐴)
4443ex 401 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵) → 𝐵 ∈ suc 𝐴))
4515, 44jcad 508 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵) → (𝐴𝐵𝐵 ∈ suc 𝐴)))
46453adant3 1162 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2𝑜 ·𝑜 𝐴)) → (suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵) → (𝐴𝐵𝐵 ∈ suc 𝐴)))
475, 46sylbid 231 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2𝑜 ·𝑜 𝐴)) → (suc 𝐶 = (2𝑜 ·𝑜 𝐵) → (𝐴𝐵𝐵 ∈ suc 𝐴)))
482, 47mtod 189 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2𝑜 ·𝑜 𝐴)) → ¬ suc 𝐶 = (2𝑜 ·𝑜 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  c0 4081  Oncon0 5910  suc csuc 5912  (class class class)co 6846  1𝑜c1o 7761  2𝑜c2o 7762   +𝑜 coa 7765   ·𝑜 comu 7766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-om 7268  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-1o 7768  df-2o 7769  df-oadd 7772  df-omul 7773
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator