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Theorem oeord 7613
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 7612 . . 3 ((𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴𝐵 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵)))
213adant1 1077 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴𝐵 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵)))
3 oveq2 6612 . . . . . 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 7612 . . . . . 6 ((𝐴 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐵𝐴 → (𝐶𝑜 𝐵) ∈ (𝐶𝑜 𝐴)))
653adant2 1078 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐵𝐴 → (𝐶𝑜 𝐵) ∈ (𝐶𝑜 𝐴)))
74, 6orim12d 882 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → ((𝐴 = 𝐵𝐵𝐴) → ((𝐶𝑜 𝐴) = (𝐶𝑜 𝐵) ∨ (𝐶𝑜 𝐵) ∈ (𝐶𝑜 𝐴))))
87con3d 148 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (¬ ((𝐶𝑜 𝐴) = (𝐶𝑜 𝐵) ∨ (𝐶𝑜 𝐵) ∈ (𝐶𝑜 𝐴)) → ¬ (𝐴 = 𝐵𝐵𝐴)))
9 eldifi 3710 . . . . . 6 (𝐶 ∈ (On ∖ 2𝑜) → 𝐶 ∈ On)
1093ad2ant3 1082 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → 𝐶 ∈ On)
11 simp1 1059 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → 𝐴 ∈ On)
12 oecl 7562 . . . . 5 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶𝑜 𝐴) ∈ On)
1310, 11, 12syl2anc 692 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐶𝑜 𝐴) ∈ On)
14 simp2 1060 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → 𝐵 ∈ On)
15 oecl 7562 . . . . 5 ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶𝑜 𝐵) ∈ On)
1610, 14, 15syl2anc 692 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐶𝑜 𝐵) ∈ On)
17 eloni 5692 . . . . 5 ((𝐶𝑜 𝐴) ∈ On → Ord (𝐶𝑜 𝐴))
18 eloni 5692 . . . . 5 ((𝐶𝑜 𝐵) ∈ On → Ord (𝐶𝑜 𝐵))
19 ordtri2 5717 . . . . 5 ((Ord (𝐶𝑜 𝐴) ∧ Ord (𝐶𝑜 𝐵)) → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵) ↔ ¬ ((𝐶𝑜 𝐴) = (𝐶𝑜 𝐵) ∨ (𝐶𝑜 𝐵) ∈ (𝐶𝑜 𝐴))))
2017, 18, 19syl2an 494 . . . 4 (((𝐶𝑜 𝐴) ∈ On ∧ (𝐶𝑜 𝐵) ∈ On) → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵) ↔ ¬ ((𝐶𝑜 𝐴) = (𝐶𝑜 𝐵) ∨ (𝐶𝑜 𝐵) ∈ (𝐶𝑜 𝐴))))
2113, 16, 20syl2anc 692 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵) ↔ ¬ ((𝐶𝑜 𝐴) = (𝐶𝑜 𝐵) ∨ (𝐶𝑜 𝐵) ∈ (𝐶𝑜 𝐴))))
22 eloni 5692 . . . . 5 (𝐴 ∈ On → Ord 𝐴)
23 eloni 5692 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
24 ordtri2 5717 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
2522, 23, 24syl2an 494 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
26253adant3 1079 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
278, 21, 263imtr4d 283 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵) → 𝐴𝐵))
282, 27impbid 202 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 196  wo 383  w3a 1036   = wceq 1480  wcel 1987  cdif 3552  Ord word 5681  Oncon0 5682  (class class class)co 6604  2𝑜c2o 7499  𝑜 coe 7504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-omul 7510  df-oexp 7511
This theorem is referenced by:  oeword  7615  oeeui  7627  omabs  7672  cantnflem3  8532
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