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Theorem oeord 7555
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 7554 . . 3 ((𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴𝐵 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵)))
213adant1 1072 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴𝐵 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵)))
3 oveq2 6557 . . . . . 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 7554 . . . . . 6 ((𝐴 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐵𝐴 → (𝐶𝑜 𝐵) ∈ (𝐶𝑜 𝐴)))
653adant2 1073 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐵𝐴 → (𝐶𝑜 𝐵) ∈ (𝐶𝑜 𝐴)))
74, 6orim12d 879 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → ((𝐴 = 𝐵𝐵𝐴) → ((𝐶𝑜 𝐴) = (𝐶𝑜 𝐵) ∨ (𝐶𝑜 𝐵) ∈ (𝐶𝑜 𝐴))))
87con3d 147 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (¬ ((𝐶𝑜 𝐴) = (𝐶𝑜 𝐵) ∨ (𝐶𝑜 𝐵) ∈ (𝐶𝑜 𝐴)) → ¬ (𝐴 = 𝐵𝐵𝐴)))
9 eldifi 3694 . . . . . 6 (𝐶 ∈ (On ∖ 2𝑜) → 𝐶 ∈ On)
1093ad2ant3 1077 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → 𝐶 ∈ On)
11 simp1 1054 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → 𝐴 ∈ On)
12 oecl 7504 . . . . 5 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶𝑜 𝐴) ∈ On)
1310, 11, 12syl2anc 691 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐶𝑜 𝐴) ∈ On)
14 simp2 1055 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → 𝐵 ∈ On)
15 oecl 7504 . . . . 5 ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶𝑜 𝐵) ∈ On)
1610, 14, 15syl2anc 691 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐶𝑜 𝐵) ∈ On)
17 eloni 5650 . . . . 5 ((𝐶𝑜 𝐴) ∈ On → Ord (𝐶𝑜 𝐴))
18 eloni 5650 . . . . 5 ((𝐶𝑜 𝐵) ∈ On → Ord (𝐶𝑜 𝐵))
19 ordtri2 5675 . . . . 5 ((Ord (𝐶𝑜 𝐴) ∧ Ord (𝐶𝑜 𝐵)) → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵) ↔ ¬ ((𝐶𝑜 𝐴) = (𝐶𝑜 𝐵) ∨ (𝐶𝑜 𝐵) ∈ (𝐶𝑜 𝐴))))
2017, 18, 19syl2an 493 . . . 4 (((𝐶𝑜 𝐴) ∈ On ∧ (𝐶𝑜 𝐵) ∈ On) → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵) ↔ ¬ ((𝐶𝑜 𝐴) = (𝐶𝑜 𝐵) ∨ (𝐶𝑜 𝐵) ∈ (𝐶𝑜 𝐴))))
2113, 16, 20syl2anc 691 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵) ↔ ¬ ((𝐶𝑜 𝐴) = (𝐶𝑜 𝐵) ∨ (𝐶𝑜 𝐵) ∈ (𝐶𝑜 𝐴))))
22 eloni 5650 . . . . 5 (𝐴 ∈ On → Ord 𝐴)
23 eloni 5650 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
24 ordtri2 5675 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
2522, 23, 24syl2an 493 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
26253adant3 1074 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
278, 21, 263imtr4d 282 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵) → 𝐴𝐵))
282, 27impbid 201 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 195  wo 382  w3a 1031   = wceq 1475  wcel 1977  cdif 3537  Ord word 5639  Oncon0 5640  (class class class)co 6549  2𝑜c2o 7441  𝑜 coe 7446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-oexp 7453
This theorem is referenced by:  oeword  7557  oeeui  7569  omabs  7614  cantnflem3  8471
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