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Theorem oneo 7548
 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 5735 . . 3 (𝐴 ∈ On → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))
213ad2ant1 1075 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2𝑜 ·𝑜 𝐴)) → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))
3 suceq 5707 . . . . 5 (𝐶 = (2𝑜 ·𝑜 𝐴) → suc 𝐶 = suc (2𝑜 ·𝑜 𝐴))
43eqeq1d 2612 . . . 4 (𝐶 = (2𝑜 ·𝑜 𝐴) → (suc 𝐶 = (2𝑜 ·𝑜 𝐵) ↔ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)))
543ad2ant3 1077 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2𝑜 ·𝑜 𝐴)) → (suc 𝐶 = (2𝑜 ·𝑜 𝐵) ↔ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)))
6 ovex 6577 . . . . . . . 8 (2𝑜 ·𝑜 𝐴) ∈ V
76sucid 5721 . . . . . . 7 (2𝑜 ·𝑜 𝐴) ∈ suc (2𝑜 ·𝑜 𝐴)
8 eleq2 2677 . . . . . . 7 (suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵) → ((2𝑜 ·𝑜 𝐴) ∈ suc (2𝑜 ·𝑜 𝐴) ↔ (2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 𝐵)))
97, 8mpbii 222 . . . . . 6 (suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵) → (2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 𝐵))
10 2on 7455 . . . . . . . 8 2𝑜 ∈ On
11 omord 7535 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 2𝑜 ∈ On) → ((𝐴𝐵 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 𝐵)))
1210, 11mp3an3 1405 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴𝐵 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 𝐵)))
13 simpl 472 . . . . . . 7 ((𝐴𝐵 ∧ ∅ ∈ 2𝑜) → 𝐴𝐵)
1412, 13syl6bir 243 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 𝐵) → 𝐴𝐵))
159, 14syl5 33 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵) → 𝐴𝐵))
16 simpr 476 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)) → suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵))
17 omcl 7503 . . . . . . . . . . . . 13 ((2𝑜 ∈ On ∧ 𝐴 ∈ On) → (2𝑜 ·𝑜 𝐴) ∈ On)
1810, 17mpan 702 . . . . . . . . . . . 12 (𝐴 ∈ On → (2𝑜 ·𝑜 𝐴) ∈ On)
19 oa1suc 7498 . . . . . . . . . . . 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 7454 . . . . . . . . . . . . . . . 16 1𝑜 ∈ On
2221elexi 3186 . . . . . . . . . . . . . . 15 1𝑜 ∈ V
2322sucid 5721 . . . . . . . . . . . . . 14 1𝑜 ∈ suc 1𝑜
24 df-2o 7448 . . . . . . . . . . . . . 14 2𝑜 = suc 1𝑜
2523, 24eleqtrri 2687 . . . . . . . . . . . . 13 1𝑜 ∈ 2𝑜
26 oaord 7514 . . . . . . . . . . . . . . 15 ((1𝑜 ∈ On ∧ 2𝑜 ∈ On ∧ (2𝑜 ·𝑜 𝐴) ∈ On) → (1𝑜 ∈ 2𝑜 ↔ ((2𝑜 ·𝑜 𝐴) +𝑜 1𝑜) ∈ ((2𝑜 ·𝑜 𝐴) +𝑜 2𝑜)))
2721, 10, 26mp3an12 1406 . . . . . . . . . . . . . 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 222 . . . . . . . . . . . 12 (𝐴 ∈ On → ((2𝑜 ·𝑜 𝐴) +𝑜 1𝑜) ∈ ((2𝑜 ·𝑜 𝐴) +𝑜 2𝑜))
30 omsuc 7493 . . . . . . . . . . . . 13 ((2𝑜 ∈ On ∧ 𝐴 ∈ On) → (2𝑜 ·𝑜 suc 𝐴) = ((2𝑜 ·𝑜 𝐴) +𝑜 2𝑜))
3110, 30mpan 702 . . . . . . . . . . . 12 (𝐴 ∈ On → (2𝑜 ·𝑜 suc 𝐴) = ((2𝑜 ·𝑜 𝐴) +𝑜 2𝑜))
3229, 31eleqtrrd 2691 . . . . . . . . . . 11 (𝐴 ∈ On → ((2𝑜 ·𝑜 𝐴) +𝑜 1𝑜) ∈ (2𝑜 ·𝑜 suc 𝐴))
3320, 32eqeltrrd 2689 . . . . . . . . . 10 (𝐴 ∈ On → suc (2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 suc 𝐴))
3433ad2antrr 758 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)) → suc (2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 suc 𝐴))
3516, 34eqeltrrd 2689 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)) → (2𝑜 ·𝑜 𝐵) ∈ (2𝑜 ·𝑜 suc 𝐴))
36 suceloni 6905 . . . . . . . . . . 11 (𝐴 ∈ On → suc 𝐴 ∈ On)
37 omord 7535 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ suc 𝐴 ∈ On ∧ 2𝑜 ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐵) ∈ (2𝑜 ·𝑜 suc 𝐴)))
3810, 37mp3an3 1405 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ suc 𝐴 ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐵) ∈ (2𝑜 ·𝑜 suc 𝐴)))
3936, 38sylan2 490 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐵) ∈ (2𝑜 ·𝑜 suc 𝐴)))
4039ancoms 468 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐵) ∈ (2𝑜 ·𝑜 suc 𝐴)))
4140adantr 480 . . . . . . . 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 246 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)) → (𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2𝑜))
4342simpld 474 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)) → 𝐵 ∈ suc 𝐴)
4443ex 449 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵) → 𝐵 ∈ suc 𝐴))
4515, 44jcad 554 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵) → (𝐴𝐵𝐵 ∈ suc 𝐴)))
46453adant3 1074 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2𝑜 ·𝑜 𝐴)) → (suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵) → (𝐴𝐵𝐵 ∈ suc 𝐴)))
475, 46sylbid 229 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2𝑜 ·𝑜 𝐴)) → (suc 𝐶 = (2𝑜 ·𝑜 𝐵) → (𝐴𝐵𝐵 ∈ suc 𝐴)))
482, 47mtod 188 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∅c0 3874  Oncon0 5640  suc csuc 5642  (class class class)co 6549  1𝑜c1o 7440  2𝑜c2o 7441   +𝑜 coa 7444   ·𝑜 comu 7445 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452 This theorem is referenced by: (None)
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