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Theorem oecan 8204
Description: Left cancellation law for ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)
Assertion
Ref Expression
oecan ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴o 𝐵) = (𝐴o 𝐶) ↔ 𝐵 = 𝐶))

Proof of Theorem oecan
StepHypRef Expression
1 oeordi 8202 . . . . . . 7 ((𝐶 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝐵𝐶 → (𝐴o 𝐵) ∈ (𝐴o 𝐶)))
21ancoms 459 . . . . . 6 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ On) → (𝐵𝐶 → (𝐴o 𝐵) ∈ (𝐴o 𝐶)))
323adant2 1123 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝐶 → (𝐴o 𝐵) ∈ (𝐴o 𝐶)))
4 oeordi 8202 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝐶𝐵 → (𝐴o 𝐶) ∈ (𝐴o 𝐵)))
54ancoms 459 . . . . . 6 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → (𝐶𝐵 → (𝐴o 𝐶) ∈ (𝐴o 𝐵)))
653adant3 1124 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶𝐵 → (𝐴o 𝐶) ∈ (𝐴o 𝐵)))
73, 6orim12d 958 . . . 4 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵𝐶𝐶𝐵) → ((𝐴o 𝐵) ∈ (𝐴o 𝐶) ∨ (𝐴o 𝐶) ∈ (𝐴o 𝐵))))
87con3d 155 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ ((𝐴o 𝐵) ∈ (𝐴o 𝐶) ∨ (𝐴o 𝐶) ∈ (𝐴o 𝐵)) → ¬ (𝐵𝐶𝐶𝐵)))
9 eldifi 4100 . . . . . 6 (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
1093ad2ant1 1125 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐴 ∈ On)
11 simp2 1129 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐵 ∈ On)
12 oecl 8151 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
1310, 11, 12syl2anc 584 . . . 4 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴o 𝐵) ∈ On)
14 simp3 1130 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐶 ∈ On)
15 oecl 8151 . . . . 5 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴o 𝐶) ∈ On)
1610, 14, 15syl2anc 584 . . . 4 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴o 𝐶) ∈ On)
17 eloni 6194 . . . . 5 ((𝐴o 𝐵) ∈ On → Ord (𝐴o 𝐵))
18 eloni 6194 . . . . 5 ((𝐴o 𝐶) ∈ On → Ord (𝐴o 𝐶))
19 ordtri3 6220 . . . . 5 ((Ord (𝐴o 𝐵) ∧ Ord (𝐴o 𝐶)) → ((𝐴o 𝐵) = (𝐴o 𝐶) ↔ ¬ ((𝐴o 𝐵) ∈ (𝐴o 𝐶) ∨ (𝐴o 𝐶) ∈ (𝐴o 𝐵))))
2017, 18, 19syl2an 595 . . . 4 (((𝐴o 𝐵) ∈ On ∧ (𝐴o 𝐶) ∈ On) → ((𝐴o 𝐵) = (𝐴o 𝐶) ↔ ¬ ((𝐴o 𝐵) ∈ (𝐴o 𝐶) ∨ (𝐴o 𝐶) ∈ (𝐴o 𝐵))))
2113, 16, 20syl2anc 584 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴o 𝐵) = (𝐴o 𝐶) ↔ ¬ ((𝐴o 𝐵) ∈ (𝐴o 𝐶) ∨ (𝐴o 𝐶) ∈ (𝐴o 𝐵))))
22 eloni 6194 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
23 eloni 6194 . . . . 5 (𝐶 ∈ On → Ord 𝐶)
24 ordtri3 6220 . . . . 5 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
2522, 23, 24syl2an 595 . . . 4 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
26253adant1 1122 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
278, 21, 263imtr4d 295 . 2 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴o 𝐵) = (𝐴o 𝐶) → 𝐵 = 𝐶))
28 oveq2 7153 . 2 (𝐵 = 𝐶 → (𝐴o 𝐵) = (𝐴o 𝐶))
2927, 28impbid1 226 1 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴o 𝐵) = (𝐴o 𝐶) ↔ 𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wo 841  w3a 1079   = wceq 1528  wcel 2105  cdif 3930  Ord word 6183  Oncon0 6184  (class class class)co 7145  2oc2o 8085  o coe 8090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-omul 8096  df-oexp 8097
This theorem is referenced by:  oeword  8205  infxpenc2lem1  9433
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