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Theorem oeord 8608
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 8607 . . 3 ((𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 → (𝐶o 𝐴) ∈ (𝐶o 𝐵)))
213adant1 1128 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 → (𝐶o 𝐴) ∈ (𝐶o 𝐵)))
3 oveq2 7428 . . . . . 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 8607 . . . . . 6 ((𝐴 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐵𝐴 → (𝐶o 𝐵) ∈ (𝐶o 𝐴)))
653adant2 1129 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐵𝐴 → (𝐶o 𝐵) ∈ (𝐶o 𝐴)))
74, 6orim12d 963 . . . 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 4125 . . . . . 6 (𝐶 ∈ (On ∖ 2o) → 𝐶 ∈ On)
1093ad2ant3 1133 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → 𝐶 ∈ On)
11 simp1 1134 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → 𝐴 ∈ On)
12 oecl 8557 . . . . 5 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶o 𝐴) ∈ On)
1310, 11, 12syl2anc 583 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐶o 𝐴) ∈ On)
14 simp2 1135 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → 𝐵 ∈ On)
15 oecl 8557 . . . . 5 ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶o 𝐵) ∈ On)
1610, 14, 15syl2anc 583 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐶o 𝐵) ∈ On)
17 eloni 6379 . . . . 5 ((𝐶o 𝐴) ∈ On → Ord (𝐶o 𝐴))
18 eloni 6379 . . . . 5 ((𝐶o 𝐵) ∈ On → Ord (𝐶o 𝐵))
19 ordtri2 6404 . . . . 5 ((Ord (𝐶o 𝐴) ∧ Ord (𝐶o 𝐵)) → ((𝐶o 𝐴) ∈ (𝐶o 𝐵) ↔ ¬ ((𝐶o 𝐴) = (𝐶o 𝐵) ∨ (𝐶o 𝐵) ∈ (𝐶o 𝐴))))
2017, 18, 19syl2an 595 . . . 4 (((𝐶o 𝐴) ∈ On ∧ (𝐶o 𝐵) ∈ On) → ((𝐶o 𝐴) ∈ (𝐶o 𝐵) ↔ ¬ ((𝐶o 𝐴) = (𝐶o 𝐵) ∨ (𝐶o 𝐵) ∈ (𝐶o 𝐴))))
2113, 16, 20syl2anc 583 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐶o 𝐴) ∈ (𝐶o 𝐵) ↔ ¬ ((𝐶o 𝐴) = (𝐶o 𝐵) ∨ (𝐶o 𝐵) ∈ (𝐶o 𝐴))))
22 eloni 6379 . . . . 5 (𝐴 ∈ On → Ord 𝐴)
23 eloni 6379 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
24 ordtri2 6404 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
2522, 23, 24syl2an 595 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
26253adant3 1130 . . 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 846  w3a 1085   = wceq 1534  wcel 2099  cdif 3944  Ord word 6368  Oncon0 6369  (class class class)co 7420  2oc2o 8480  o coe 8485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-1o 8486  df-2o 8487  df-oadd 8490  df-omul 8491  df-oexp 8492
This theorem is referenced by:  oeword  8610  oeeui  8622  omabs  8671  cantnflem3  9714  oeord2com  42740  omabs2  42761
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