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Theorem oneo 8190
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 6250 . . 3 (𝐴 ∈ On → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))
213ad2ant1 1130 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2o ·o 𝐴)) → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))
3 suceq 6224 . . . . 5 (𝐶 = (2o ·o 𝐴) → suc 𝐶 = suc (2o ·o 𝐴))
43eqeq1d 2800 . . . 4 (𝐶 = (2o ·o 𝐴) → (suc 𝐶 = (2o ·o 𝐵) ↔ suc (2o ·o 𝐴) = (2o ·o 𝐵)))
543ad2ant3 1132 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2o ·o 𝐴)) → (suc 𝐶 = (2o ·o 𝐵) ↔ suc (2o ·o 𝐴) = (2o ·o 𝐵)))
6 ovex 7168 . . . . . . . 8 (2o ·o 𝐴) ∈ V
76sucid 6238 . . . . . . 7 (2o ·o 𝐴) ∈ suc (2o ·o 𝐴)
8 eleq2 2878 . . . . . . 7 (suc (2o ·o 𝐴) = (2o ·o 𝐵) → ((2o ·o 𝐴) ∈ suc (2o ·o 𝐴) ↔ (2o ·o 𝐴) ∈ (2o ·o 𝐵)))
97, 8mpbii 236 . . . . . 6 (suc (2o ·o 𝐴) = (2o ·o 𝐵) → (2o ·o 𝐴) ∈ (2o ·o 𝐵))
10 2on 8094 . . . . . . . 8 2o ∈ On
11 omord 8177 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 2o ∈ On) → ((𝐴𝐵 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐴) ∈ (2o ·o 𝐵)))
1210, 11mp3an3 1447 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴𝐵 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐴) ∈ (2o ·o 𝐵)))
13 simpl 486 . . . . . . 7 ((𝐴𝐵 ∧ ∅ ∈ 2o) → 𝐴𝐵)
1412, 13syl6bir 257 . . . . . 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 488 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → suc (2o ·o 𝐴) = (2o ·o 𝐵))
17 omcl 8144 . . . . . . . . . . . . 13 ((2o ∈ On ∧ 𝐴 ∈ On) → (2o ·o 𝐴) ∈ On)
1810, 17mpan 689 . . . . . . . . . . . 12 (𝐴 ∈ On → (2o ·o 𝐴) ∈ On)
19 oa1suc 8139 . . . . . . . . . . . 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 8093 . . . . . . . . . . . . . . 15 1o ∈ V
2221sucid 6238 . . . . . . . . . . . . . 14 1o ∈ suc 1o
23 df-2o 8086 . . . . . . . . . . . . . 14 2o = suc 1o
2422, 23eleqtrri 2889 . . . . . . . . . . . . 13 1o ∈ 2o
25 1on 8092 . . . . . . . . . . . . . 14 1o ∈ On
26 oaord 8156 . . . . . . . . . . . . . 14 ((1o ∈ On ∧ 2o ∈ On ∧ (2o ·o 𝐴) ∈ On) → (1o ∈ 2o ↔ ((2o ·o 𝐴) +o 1o) ∈ ((2o ·o 𝐴) +o 2o)))
2725, 10, 18, 26mp3an12i 1462 . . . . . . . . . . . . 13 (𝐴 ∈ On → (1o ∈ 2o ↔ ((2o ·o 𝐴) +o 1o) ∈ ((2o ·o 𝐴) +o 2o)))
2824, 27mpbii 236 . . . . . . . . . . . 12 (𝐴 ∈ On → ((2o ·o 𝐴) +o 1o) ∈ ((2o ·o 𝐴) +o 2o))
29 omsuc 8134 . . . . . . . . . . . . 13 ((2o ∈ On ∧ 𝐴 ∈ On) → (2o ·o suc 𝐴) = ((2o ·o 𝐴) +o 2o))
3010, 29mpan 689 . . . . . . . . . . . 12 (𝐴 ∈ On → (2o ·o suc 𝐴) = ((2o ·o 𝐴) +o 2o))
3128, 30eleqtrrd 2893 . . . . . . . . . . 11 (𝐴 ∈ On → ((2o ·o 𝐴) +o 1o) ∈ (2o ·o suc 𝐴))
3220, 31eqeltrrd 2891 . . . . . . . . . 10 (𝐴 ∈ On → suc (2o ·o 𝐴) ∈ (2o ·o suc 𝐴))
3332ad2antrr 725 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → suc (2o ·o 𝐴) ∈ (2o ·o suc 𝐴))
3416, 33eqeltrrd 2891 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → (2o ·o 𝐵) ∈ (2o ·o suc 𝐴))
35 suceloni 7508 . . . . . . . . . . 11 (𝐴 ∈ On → suc 𝐴 ∈ On)
36 omord 8177 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ suc 𝐴 ∈ On ∧ 2o ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
3710, 36mp3an3 1447 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ suc 𝐴 ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
3835, 37sylan2 595 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
3938ancoms 462 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
4039adantr 484 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
4134, 40mpbird 260 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → (𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o))
4241simpld 498 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → 𝐵 ∈ suc 𝐴)
4342ex 416 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc (2o ·o 𝐴) = (2o ·o 𝐵) → 𝐵 ∈ suc 𝐴))
4415, 43jcad 516 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc (2o ·o 𝐴) = (2o ·o 𝐵) → (𝐴𝐵𝐵 ∈ suc 𝐴)))
45443adant3 1129 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2o ·o 𝐴)) → (suc (2o ·o 𝐴) = (2o ·o 𝐵) → (𝐴𝐵𝐵 ∈ suc 𝐴)))
465, 45sylbid 243 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2o ·o 𝐴)) → (suc 𝐶 = (2o ·o 𝐵) → (𝐴𝐵𝐵 ∈ suc 𝐴)))
472, 46mtod 201 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2111  c0 4243  Oncon0 6159  suc csuc 6161  (class class class)co 7135  1oc1o 8078  2oc2o 8079   +o coa 8082   ·o comu 8083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-omul 8090
This theorem is referenced by: (None)
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