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Theorem oeord 8402
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 8401 . . 3 ((𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 → (𝐶o 𝐴) ∈ (𝐶o 𝐵)))
213adant1 1129 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 → (𝐶o 𝐴) ∈ (𝐶o 𝐵)))
3 oveq2 7277 . . . . . 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 8401 . . . . . 6 ((𝐴 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐵𝐴 → (𝐶o 𝐵) ∈ (𝐶o 𝐴)))
653adant2 1130 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐵𝐴 → (𝐶o 𝐵) ∈ (𝐶o 𝐴)))
74, 6orim12d 962 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐴 = 𝐵𝐵𝐴) → ((𝐶o 𝐴) = (𝐶o 𝐵) ∨ (𝐶o 𝐵) ∈ (𝐶o 𝐴))))
87con3d 152 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (¬ ((𝐶o 𝐴) = (𝐶o 𝐵) ∨ (𝐶o 𝐵) ∈ (𝐶o 𝐴)) → ¬ (𝐴 = 𝐵𝐵𝐴)))
9 eldifi 4066 . . . . . 6 (𝐶 ∈ (On ∖ 2o) → 𝐶 ∈ On)
1093ad2ant3 1134 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → 𝐶 ∈ On)
11 simp1 1135 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → 𝐴 ∈ On)
12 oecl 8350 . . . . 5 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶o 𝐴) ∈ On)
1310, 11, 12syl2anc 584 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐶o 𝐴) ∈ On)
14 simp2 1136 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → 𝐵 ∈ On)
15 oecl 8350 . . . . 5 ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶o 𝐵) ∈ On)
1610, 14, 15syl2anc 584 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐶o 𝐵) ∈ On)
17 eloni 6274 . . . . 5 ((𝐶o 𝐴) ∈ On → Ord (𝐶o 𝐴))
18 eloni 6274 . . . . 5 ((𝐶o 𝐵) ∈ On → Ord (𝐶o 𝐵))
19 ordtri2 6299 . . . . 5 ((Ord (𝐶o 𝐴) ∧ Ord (𝐶o 𝐵)) → ((𝐶o 𝐴) ∈ (𝐶o 𝐵) ↔ ¬ ((𝐶o 𝐴) = (𝐶o 𝐵) ∨ (𝐶o 𝐵) ∈ (𝐶o 𝐴))))
2017, 18, 19syl2an 596 . . . 4 (((𝐶o 𝐴) ∈ On ∧ (𝐶o 𝐵) ∈ On) → ((𝐶o 𝐴) ∈ (𝐶o 𝐵) ↔ ¬ ((𝐶o 𝐴) = (𝐶o 𝐵) ∨ (𝐶o 𝐵) ∈ (𝐶o 𝐴))))
2113, 16, 20syl2anc 584 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐶o 𝐴) ∈ (𝐶o 𝐵) ↔ ¬ ((𝐶o 𝐴) = (𝐶o 𝐵) ∨ (𝐶o 𝐵) ∈ (𝐶o 𝐴))))
22 eloni 6274 . . . . 5 (𝐴 ∈ On → Ord 𝐴)
23 eloni 6274 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
24 ordtri2 6299 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
2522, 23, 24syl2an 596 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
26253adant3 1131 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
278, 21, 263imtr4d 294 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐶o 𝐴) ∈ (𝐶o 𝐵) → 𝐴𝐵))
282, 27impbid 211 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 205  wo 844  w3a 1086   = wceq 1542  wcel 2110  cdif 3889  Ord word 6263  Oncon0 6264  (class class class)co 7269  2oc2o 8280  o coe 8285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7580
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6200  df-ord 6267  df-on 6268  df-lim 6269  df-suc 6270  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-f1 6436  df-fo 6437  df-f1o 6438  df-fv 6439  df-ov 7272  df-oprab 7273  df-mpo 7274  df-om 7705  df-2nd 7823  df-frecs 8086  df-wrecs 8117  df-recs 8191  df-rdg 8230  df-1o 8286  df-2o 8287  df-oadd 8290  df-omul 8291  df-oexp 8292
This theorem is referenced by:  oeword  8404  oeeui  8416  omabs  8462  cantnflem3  9425
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