MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oeord Structured version   Visualization version   GIF version

Theorem oeord 8064
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 8063 . . 3 ((𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 → (𝐶o 𝐴) ∈ (𝐶o 𝐵)))
213adant1 1123 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 → (𝐶o 𝐴) ∈ (𝐶o 𝐵)))
3 oveq2 7024 . . . . . 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 8063 . . . . . 6 ((𝐴 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐵𝐴 → (𝐶o 𝐵) ∈ (𝐶o 𝐴)))
653adant2 1124 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐵𝐴 → (𝐶o 𝐵) ∈ (𝐶o 𝐴)))
74, 6orim12d 959 . . . 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 4024 . . . . . 6 (𝐶 ∈ (On ∖ 2o) → 𝐶 ∈ On)
1093ad2ant3 1128 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → 𝐶 ∈ On)
11 simp1 1129 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → 𝐴 ∈ On)
12 oecl 8013 . . . . 5 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶o 𝐴) ∈ On)
1310, 11, 12syl2anc 584 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐶o 𝐴) ∈ On)
14 simp2 1130 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → 𝐵 ∈ On)
15 oecl 8013 . . . . 5 ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶o 𝐵) ∈ On)
1610, 14, 15syl2anc 584 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐶o 𝐵) ∈ On)
17 eloni 6076 . . . . 5 ((𝐶o 𝐴) ∈ On → Ord (𝐶o 𝐴))
18 eloni 6076 . . . . 5 ((𝐶o 𝐵) ∈ On → Ord (𝐶o 𝐵))
19 ordtri2 6101 . . . . 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 584 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐶o 𝐴) ∈ (𝐶o 𝐵) ↔ ¬ ((𝐶o 𝐴) = (𝐶o 𝐵) ∨ (𝐶o 𝐵) ∈ (𝐶o 𝐴))))
22 eloni 6076 . . . . 5 (𝐴 ∈ On → Ord 𝐴)
23 eloni 6076 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
24 ordtri2 6101 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
2522, 23, 24syl2an 595 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
26253adant3 1125 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
278, 21, 263imtr4d 295 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐶o 𝐴) ∈ (𝐶o 𝐵) → 𝐴𝐵))
282, 27impbid 213 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 207  wo 842  w3a 1080   = wceq 1522  wcel 2081  cdif 3856  Ord word 6065  Oncon0 6066  (class class class)co 7016  2oc2o 7947  o coe 7952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-2o 7954  df-oadd 7957  df-omul 7958  df-oexp 7959
This theorem is referenced by:  oeword  8066  oeeui  8078  omabs  8124  cantnflem3  9000
  Copyright terms: Public domain W3C validator