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Theorem oeord 7788
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 7787 . . 3 ((𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴𝐵 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵)))
213adant1 1122 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴𝐵 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵)))
3 oveq2 6773 . . . . . 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 7787 . . . . . 6 ((𝐴 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐵𝐴 → (𝐶𝑜 𝐵) ∈ (𝐶𝑜 𝐴)))
653adant2 1123 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐵𝐴 → (𝐶𝑜 𝐵) ∈ (𝐶𝑜 𝐴)))
74, 6orim12d 919 . . . 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 3840 . . . . . 6 (𝐶 ∈ (On ∖ 2𝑜) → 𝐶 ∈ On)
1093ad2ant3 1127 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → 𝐶 ∈ On)
11 simp1 1128 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → 𝐴 ∈ On)
12 oecl 7737 . . . . 5 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶𝑜 𝐴) ∈ On)
1310, 11, 12syl2anc 696 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐶𝑜 𝐴) ∈ On)
14 simp2 1129 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → 𝐵 ∈ On)
15 oecl 7737 . . . . 5 ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶𝑜 𝐵) ∈ On)
1610, 14, 15syl2anc 696 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐶𝑜 𝐵) ∈ On)
17 eloni 5846 . . . . 5 ((𝐶𝑜 𝐴) ∈ On → Ord (𝐶𝑜 𝐴))
18 eloni 5846 . . . . 5 ((𝐶𝑜 𝐵) ∈ On → Ord (𝐶𝑜 𝐵))
19 ordtri2 5871 . . . . 5 ((Ord (𝐶𝑜 𝐴) ∧ Ord (𝐶𝑜 𝐵)) → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵) ↔ ¬ ((𝐶𝑜 𝐴) = (𝐶𝑜 𝐵) ∨ (𝐶𝑜 𝐵) ∈ (𝐶𝑜 𝐴))))
2017, 18, 19syl2an 495 . . . 4 (((𝐶𝑜 𝐴) ∈ On ∧ (𝐶𝑜 𝐵) ∈ On) → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵) ↔ ¬ ((𝐶𝑜 𝐴) = (𝐶𝑜 𝐵) ∨ (𝐶𝑜 𝐵) ∈ (𝐶𝑜 𝐴))))
2113, 16, 20syl2anc 696 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → ((𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵) ↔ ¬ ((𝐶𝑜 𝐴) = (𝐶𝑜 𝐵) ∨ (𝐶𝑜 𝐵) ∈ (𝐶𝑜 𝐴))))
22 eloni 5846 . . . . 5 (𝐴 ∈ On → Ord 𝐴)
23 eloni 5846 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
24 ordtri2 5871 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
2522, 23, 24syl2an 495 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
26253adant3 1124 . . 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 382  w3a 1072   = wceq 1596  wcel 2103  cdif 3677  Ord word 5835  Oncon0 5836  (class class class)co 6765  2𝑜c2o 7674  𝑜 coe 7679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-2o 7681  df-oadd 7684  df-omul 7685  df-oexp 7686
This theorem is referenced by:  oeword  7790  oeeui  7802  omabs  7847  cantnflem3  8701
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