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Theorem oeord 8197
 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 8196 . . 3 ((𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 → (𝐶o 𝐴) ∈ (𝐶o 𝐵)))
213adant1 1127 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 → (𝐶o 𝐴) ∈ (𝐶o 𝐵)))
3 oveq2 7143 . . . . . 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 8196 . . . . . 6 ((𝐴 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐵𝐴 → (𝐶o 𝐵) ∈ (𝐶o 𝐴)))
653adant2 1128 . . . . 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 155 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (¬ ((𝐶o 𝐴) = (𝐶o 𝐵) ∨ (𝐶o 𝐵) ∈ (𝐶o 𝐴)) → ¬ (𝐴 = 𝐵𝐵𝐴)))
9 eldifi 4054 . . . . . 6 (𝐶 ∈ (On ∖ 2o) → 𝐶 ∈ On)
1093ad2ant3 1132 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → 𝐶 ∈ On)
11 simp1 1133 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → 𝐴 ∈ On)
12 oecl 8145 . . . . 5 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶o 𝐴) ∈ On)
1310, 11, 12syl2anc 587 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐶o 𝐴) ∈ On)
14 simp2 1134 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → 𝐵 ∈ On)
15 oecl 8145 . . . . 5 ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶o 𝐵) ∈ On)
1610, 14, 15syl2anc 587 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐶o 𝐵) ∈ On)
17 eloni 6169 . . . . 5 ((𝐶o 𝐴) ∈ On → Ord (𝐶o 𝐴))
18 eloni 6169 . . . . 5 ((𝐶o 𝐵) ∈ On → Ord (𝐶o 𝐵))
19 ordtri2 6194 . . . . 5 ((Ord (𝐶o 𝐴) ∧ Ord (𝐶o 𝐵)) → ((𝐶o 𝐴) ∈ (𝐶o 𝐵) ↔ ¬ ((𝐶o 𝐴) = (𝐶o 𝐵) ∨ (𝐶o 𝐵) ∈ (𝐶o 𝐴))))
2017, 18, 19syl2an 598 . . . 4 (((𝐶o 𝐴) ∈ On ∧ (𝐶o 𝐵) ∈ On) → ((𝐶o 𝐴) ∈ (𝐶o 𝐵) ↔ ¬ ((𝐶o 𝐴) = (𝐶o 𝐵) ∨ (𝐶o 𝐵) ∈ (𝐶o 𝐴))))
2113, 16, 20syl2anc 587 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐶o 𝐴) ∈ (𝐶o 𝐵) ↔ ¬ ((𝐶o 𝐴) = (𝐶o 𝐵) ∨ (𝐶o 𝐵) ∈ (𝐶o 𝐴))))
22 eloni 6169 . . . . 5 (𝐴 ∈ On → Ord 𝐴)
23 eloni 6169 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
24 ordtri2 6194 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
2522, 23, 24syl2an 598 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
26253adant3 1129 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
278, 21, 263imtr4d 297 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐶o 𝐴) ∈ (𝐶o 𝐵) → 𝐴𝐵))
282, 27impbid 215 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 209   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ∖ cdif 3878  Ord word 6158  Oncon0 6159  (class class class)co 7135  2oc2o 8079   ↑o coe 8084 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  df-oexp 8091 This theorem is referenced by:  oeword  8199  oeeui  8211  omabs  8257  cantnflem3  9138
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