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Theorem oeord 7905
Description: Ordering property of ordinal exponentiation. Corollary 8.34 of [TakeutiZaring] p. 68 and its converse. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)
Assertion
Ref Expression
oeord ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴𝐵 ↔ (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵)))

Proof of Theorem oeord
StepHypRef Expression
1 oeordi 7904 . . 3 ((𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴𝐵 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵)))
213adant1 1153 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴𝐵 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵)))
3 oveq2 6882 . . . . . 6 (𝐴 = 𝐵 → (𝐶𝑜 𝐴) = (𝐶𝑜 𝐵))
43a1i 11 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴 = 𝐵 → (𝐶𝑜 𝐴) = (𝐶𝑜 𝐵)))
5 oeordi 7904 . . . . . 6 ((𝐴 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐵𝐴 → (𝐶𝑜 𝐵) ∈ (𝐶𝑜 𝐴)))
653adant2 1154 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐵𝐴 → (𝐶𝑜 𝐵) ∈ (𝐶𝑜 𝐴)))
74, 6orim12d 978 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → ((𝐴 = 𝐵𝐵𝐴) → ((𝐶𝑜 𝐴) = (𝐶𝑜 𝐵) ∨ (𝐶𝑜 𝐵) ∈ (𝐶𝑜 𝐴))))
87con3d 149 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (¬ ((𝐶𝑜 𝐴) = (𝐶𝑜 𝐵) ∨ (𝐶𝑜 𝐵) ∈ (𝐶𝑜 𝐴)) → ¬ (𝐴 = 𝐵𝐵𝐴)))
9 eldifi 3931 . . . . . 6 (𝐶 ∈ (On ∖ 2𝑜) → 𝐶 ∈ On)
1093ad2ant3 1158 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → 𝐶 ∈ On)
11 simp1 1159 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → 𝐴 ∈ On)
12 oecl 7854 . . . . 5 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶𝑜 𝐴) ∈ On)
1310, 11, 12syl2anc 575 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐶𝑜 𝐴) ∈ On)
14 simp2 1160 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → 𝐵 ∈ On)
15 oecl 7854 . . . . 5 ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶𝑜 𝐵) ∈ On)
1610, 14, 15syl2anc 575 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐶𝑜 𝐵) ∈ On)
17 eloni 5946 . . . . 5 ((𝐶𝑜 𝐴) ∈ On → Ord (𝐶𝑜 𝐴))
18 eloni 5946 . . . . 5 ((𝐶𝑜 𝐵) ∈ On → Ord (𝐶𝑜 𝐵))
19 ordtri2 5971 . . . . 5 ((Ord (𝐶𝑜 𝐴) ∧ Ord (𝐶𝑜 𝐵)) → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵) ↔ ¬ ((𝐶𝑜 𝐴) = (𝐶𝑜 𝐵) ∨ (𝐶𝑜 𝐵) ∈ (𝐶𝑜 𝐴))))
2017, 18, 19syl2an 585 . . . 4 (((𝐶𝑜 𝐴) ∈ On ∧ (𝐶𝑜 𝐵) ∈ On) → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵) ↔ ¬ ((𝐶𝑜 𝐴) = (𝐶𝑜 𝐵) ∨ (𝐶𝑜 𝐵) ∈ (𝐶𝑜 𝐴))))
2113, 16, 20syl2anc 575 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵) ↔ ¬ ((𝐶𝑜 𝐴) = (𝐶𝑜 𝐵) ∨ (𝐶𝑜 𝐵) ∈ (𝐶𝑜 𝐴))))
22 eloni 5946 . . . . 5 (𝐴 ∈ On → Ord 𝐴)
23 eloni 5946 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
24 ordtri2 5971 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
2522, 23, 24syl2an 585 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
26253adant3 1155 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
278, 21, 263imtr4d 285 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵) → 𝐴𝐵))
282, 27impbid 203 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴𝐵 ↔ (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wo 865  w3a 1100   = wceq 1637  wcel 2156  cdif 3766  Ord word 5935  Oncon0 5936  (class class class)co 6874  2𝑜c2o 7790  𝑜 coe 7795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-om 7296  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-1o 7796  df-2o 7797  df-oadd 7800  df-omul 7801  df-oexp 7802
This theorem is referenced by:  oeword  7907  oeeui  7919  omabs  7964  cantnflem3  8835
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