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Theorem oneo 8057
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 6157 . . 3 (𝐴 ∈ On → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))
213ad2ant1 1126 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2o ·o 𝐴)) → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))
3 suceq 6131 . . . . 5 (𝐶 = (2o ·o 𝐴) → suc 𝐶 = suc (2o ·o 𝐴))
43eqeq1d 2797 . . . 4 (𝐶 = (2o ·o 𝐴) → (suc 𝐶 = (2o ·o 𝐵) ↔ suc (2o ·o 𝐴) = (2o ·o 𝐵)))
543ad2ant3 1128 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2o ·o 𝐴)) → (suc 𝐶 = (2o ·o 𝐵) ↔ suc (2o ·o 𝐴) = (2o ·o 𝐵)))
6 ovex 7048 . . . . . . . 8 (2o ·o 𝐴) ∈ V
76sucid 6145 . . . . . . 7 (2o ·o 𝐴) ∈ suc (2o ·o 𝐴)
8 eleq2 2871 . . . . . . 7 (suc (2o ·o 𝐴) = (2o ·o 𝐵) → ((2o ·o 𝐴) ∈ suc (2o ·o 𝐴) ↔ (2o ·o 𝐴) ∈ (2o ·o 𝐵)))
97, 8mpbii 234 . . . . . 6 (suc (2o ·o 𝐴) = (2o ·o 𝐵) → (2o ·o 𝐴) ∈ (2o ·o 𝐵))
10 2on 7962 . . . . . . . 8 2o ∈ On
11 omord 8044 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 2o ∈ On) → ((𝐴𝐵 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐴) ∈ (2o ·o 𝐵)))
1210, 11mp3an3 1442 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴𝐵 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐴) ∈ (2o ·o 𝐵)))
13 simpl 483 . . . . . . 7 ((𝐴𝐵 ∧ ∅ ∈ 2o) → 𝐴𝐵)
1412, 13syl6bir 255 . . . . . 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 485 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → suc (2o ·o 𝐴) = (2o ·o 𝐵))
17 omcl 8012 . . . . . . . . . . . . 13 ((2o ∈ On ∧ 𝐴 ∈ On) → (2o ·o 𝐴) ∈ On)
1810, 17mpan 686 . . . . . . . . . . . 12 (𝐴 ∈ On → (2o ·o 𝐴) ∈ On)
19 oa1suc 8007 . . . . . . . . . . . 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 7961 . . . . . . . . . . . . . . 15 1o ∈ V
2221sucid 6145 . . . . . . . . . . . . . 14 1o ∈ suc 1o
23 df-2o 7954 . . . . . . . . . . . . . 14 2o = suc 1o
2422, 23eleqtrri 2882 . . . . . . . . . . . . 13 1o ∈ 2o
25 1on 7960 . . . . . . . . . . . . . 14 1o ∈ On
26 oaord 8023 . . . . . . . . . . . . . 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 1457 . . . . . . . . . . . . 13 (𝐴 ∈ On → (1o ∈ 2o ↔ ((2o ·o 𝐴) +o 1o) ∈ ((2o ·o 𝐴) +o 2o)))
2824, 27mpbii 234 . . . . . . . . . . . 12 (𝐴 ∈ On → ((2o ·o 𝐴) +o 1o) ∈ ((2o ·o 𝐴) +o 2o))
29 omsuc 8002 . . . . . . . . . . . . 13 ((2o ∈ On ∧ 𝐴 ∈ On) → (2o ·o suc 𝐴) = ((2o ·o 𝐴) +o 2o))
3010, 29mpan 686 . . . . . . . . . . . 12 (𝐴 ∈ On → (2o ·o suc 𝐴) = ((2o ·o 𝐴) +o 2o))
3128, 30eleqtrrd 2886 . . . . . . . . . . 11 (𝐴 ∈ On → ((2o ·o 𝐴) +o 1o) ∈ (2o ·o suc 𝐴))
3220, 31eqeltrrd 2884 . . . . . . . . . 10 (𝐴 ∈ On → suc (2o ·o 𝐴) ∈ (2o ·o suc 𝐴))
3332ad2antrr 722 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → suc (2o ·o 𝐴) ∈ (2o ·o suc 𝐴))
3416, 33eqeltrrd 2884 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → (2o ·o 𝐵) ∈ (2o ·o suc 𝐴))
35 suceloni 7384 . . . . . . . . . . 11 (𝐴 ∈ On → suc 𝐴 ∈ On)
36 omord 8044 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ suc 𝐴 ∈ On ∧ 2o ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
3710, 36mp3an3 1442 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ suc 𝐴 ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
3835, 37sylan2 592 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
3938ancoms 459 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
4039adantr 481 . . . . . . . 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 258 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → (𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o))
4241simpld 495 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → 𝐵 ∈ suc 𝐴)
4342ex 413 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc (2o ·o 𝐴) = (2o ·o 𝐵) → 𝐵 ∈ suc 𝐴))
4415, 43jcad 513 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc (2o ·o 𝐴) = (2o ·o 𝐵) → (𝐴𝐵𝐵 ∈ suc 𝐴)))
45443adant3 1125 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2o ·o 𝐴)) → (suc (2o ·o 𝐴) = (2o ·o 𝐵) → (𝐴𝐵𝐵 ∈ suc 𝐴)))
465, 45sylbid 241 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2o ·o 𝐴)) → (suc 𝐶 = (2o ·o 𝐵) → (𝐴𝐵𝐵 ∈ suc 𝐴)))
472, 46mtod 199 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 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  c0 4211  Oncon0 6066  suc csuc 6068  (class class class)co 7016  1oc1o 7946  2oc2o 7947   +o coa 7950   ·o comu 7951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-2o 7954  df-oadd 7957  df-omul 7958
This theorem is referenced by: (None)
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