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Theorem oneo 7609
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 5779 . . 3 (𝐴 ∈ On → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))
213ad2ant1 1080 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2𝑜 ·𝑜 𝐴)) → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))
3 suceq 5751 . . . . 5 (𝐶 = (2𝑜 ·𝑜 𝐴) → suc 𝐶 = suc (2𝑜 ·𝑜 𝐴))
43eqeq1d 2623 . . . 4 (𝐶 = (2𝑜 ·𝑜 𝐴) → (suc 𝐶 = (2𝑜 ·𝑜 𝐵) ↔ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)))
543ad2ant3 1082 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2𝑜 ·𝑜 𝐴)) → (suc 𝐶 = (2𝑜 ·𝑜 𝐵) ↔ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)))
6 ovex 6635 . . . . . . . 8 (2𝑜 ·𝑜 𝐴) ∈ V
76sucid 5765 . . . . . . 7 (2𝑜 ·𝑜 𝐴) ∈ suc (2𝑜 ·𝑜 𝐴)
8 eleq2 2687 . . . . . . 7 (suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵) → ((2𝑜 ·𝑜 𝐴) ∈ suc (2𝑜 ·𝑜 𝐴) ↔ (2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 𝐵)))
97, 8mpbii 223 . . . . . 6 (suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵) → (2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 𝐵))
10 2on 7516 . . . . . . . 8 2𝑜 ∈ On
11 omord 7596 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 2𝑜 ∈ On) → ((𝐴𝐵 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 𝐵)))
1210, 11mp3an3 1410 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴𝐵 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 𝐵)))
13 simpl 473 . . . . . . 7 ((𝐴𝐵 ∧ ∅ ∈ 2𝑜) → 𝐴𝐵)
1412, 13syl6bir 244 . . . . . 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 7564 . . . . . . . . . . . . 13 ((2𝑜 ∈ On ∧ 𝐴 ∈ On) → (2𝑜 ·𝑜 𝐴) ∈ On)
1810, 17mpan 705 . . . . . . . . . . . 12 (𝐴 ∈ On → (2𝑜 ·𝑜 𝐴) ∈ On)
19 oa1suc 7559 . . . . . . . . . . . 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 1on 7515 . . . . . . . . . . . . . . . 16 1𝑜 ∈ On
2221elexi 3199 . . . . . . . . . . . . . . 15 1𝑜 ∈ V
2322sucid 5765 . . . . . . . . . . . . . 14 1𝑜 ∈ suc 1𝑜
24 df-2o 7509 . . . . . . . . . . . . . 14 2𝑜 = suc 1𝑜
2523, 24eleqtrri 2697 . . . . . . . . . . . . 13 1𝑜 ∈ 2𝑜
26 oaord 7575 . . . . . . . . . . . . . . 15 ((1𝑜 ∈ On ∧ 2𝑜 ∈ On ∧ (2𝑜 ·𝑜 𝐴) ∈ On) → (1𝑜 ∈ 2𝑜 ↔ ((2𝑜 ·𝑜 𝐴) +𝑜 1𝑜) ∈ ((2𝑜 ·𝑜 𝐴) +𝑜 2𝑜)))
2721, 10, 26mp3an12 1411 . . . . . . . . . . . . . 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)))
2925, 28mpbii 223 . . . . . . . . . . . 12 (𝐴 ∈ On → ((2𝑜 ·𝑜 𝐴) +𝑜 1𝑜) ∈ ((2𝑜 ·𝑜 𝐴) +𝑜 2𝑜))
30 omsuc 7554 . . . . . . . . . . . . 13 ((2𝑜 ∈ On ∧ 𝐴 ∈ On) → (2𝑜 ·𝑜 suc 𝐴) = ((2𝑜 ·𝑜 𝐴) +𝑜 2𝑜))
3110, 30mpan 705 . . . . . . . . . . . 12 (𝐴 ∈ On → (2𝑜 ·𝑜 suc 𝐴) = ((2𝑜 ·𝑜 𝐴) +𝑜 2𝑜))
3229, 31eleqtrrd 2701 . . . . . . . . . . 11 (𝐴 ∈ On → ((2𝑜 ·𝑜 𝐴) +𝑜 1𝑜) ∈ (2𝑜 ·𝑜 suc 𝐴))
3320, 32eqeltrrd 2699 . . . . . . . . . 10 (𝐴 ∈ On → suc (2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 suc 𝐴))
3433ad2antrr 761 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)) → suc (2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 suc 𝐴))
3516, 34eqeltrrd 2699 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)) → (2𝑜 ·𝑜 𝐵) ∈ (2𝑜 ·𝑜 suc 𝐴))
36 suceloni 6963 . . . . . . . . . . 11 (𝐴 ∈ On → suc 𝐴 ∈ On)
37 omord 7596 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ suc 𝐴 ∈ On ∧ 2𝑜 ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐵) ∈ (2𝑜 ·𝑜 suc 𝐴)))
3810, 37mp3an3 1410 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ suc 𝐴 ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐵) ∈ (2𝑜 ·𝑜 suc 𝐴)))
3936, 38sylan2 491 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐵) ∈ (2𝑜 ·𝑜 suc 𝐴)))
4039ancoms 469 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐵) ∈ (2𝑜 ·𝑜 suc 𝐴)))
4140adantr 481 . . . . . . . 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 247 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)) → (𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2𝑜))
4342simpld 475 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)) → 𝐵 ∈ suc 𝐴)
4443ex 450 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵) → 𝐵 ∈ suc 𝐴))
4515, 44jcad 555 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵) → (𝐴𝐵𝐵 ∈ suc 𝐴)))
46453adant3 1079 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2𝑜 ·𝑜 𝐴)) → (suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵) → (𝐴𝐵𝐵 ∈ suc 𝐴)))
475, 46sylbid 230 . 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  c0 3893  Oncon0 5684  suc csuc 5686  (class class class)co 6607  1𝑜c1o 7501  2𝑜c2o 7502   +𝑜 coa 7505   ·𝑜 comu 7506
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 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
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 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-2o 7509  df-oadd 7512  df-omul 7513
This theorem is referenced by: (None)
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