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Theorem oecan 8585
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 8583 . . . . . . 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 1131 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝐶 → (𝐴o 𝐵) ∈ (𝐴o 𝐶)))
4 oeordi 8583 . . . . . . 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 1132 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶𝐵 → (𝐴o 𝐶) ∈ (𝐴o 𝐵)))
73, 6orim12d 963 . . . 4 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵𝐶𝐶𝐵) → ((𝐴o 𝐵) ∈ (𝐴o 𝐶) ∨ (𝐴o 𝐶) ∈ (𝐴o 𝐵))))
87con3d 152 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ ((𝐴o 𝐵) ∈ (𝐴o 𝐶) ∨ (𝐴o 𝐶) ∈ (𝐴o 𝐵)) → ¬ (𝐵𝐶𝐶𝐵)))
9 eldifi 4125 . . . . . 6 (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
1093ad2ant1 1133 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐴 ∈ On)
11 simp2 1137 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐵 ∈ On)
12 oecl 8533 . . . . 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 1138 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐶 ∈ On)
15 oecl 8533 . . . . 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 6371 . . . . 5 ((𝐴o 𝐵) ∈ On → Ord (𝐴o 𝐵))
18 eloni 6371 . . . . 5 ((𝐴o 𝐶) ∈ On → Ord (𝐴o 𝐶))
19 ordtri3 6397 . . . . 5 ((Ord (𝐴o 𝐵) ∧ Ord (𝐴o 𝐶)) → ((𝐴o 𝐵) = (𝐴o 𝐶) ↔ ¬ ((𝐴o 𝐵) ∈ (𝐴o 𝐶) ∨ (𝐴o 𝐶) ∈ (𝐴o 𝐵))))
2017, 18, 19syl2an 596 . . . 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 6371 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
23 eloni 6371 . . . . 5 (𝐶 ∈ On → Ord 𝐶)
24 ordtri3 6397 . . . . 5 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
2522, 23, 24syl2an 596 . . . 4 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
26253adant1 1130 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
278, 21, 263imtr4d 293 . 2 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴o 𝐵) = (𝐴o 𝐶) → 𝐵 = 𝐶))
28 oveq2 7413 . 2 (𝐵 = 𝐶 → (𝐴o 𝐵) = (𝐴o 𝐶))
2927, 28impbid1 224 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 205  wo 845  w3a 1087   = wceq 1541  wcel 2106  cdif 3944  Ord word 6360  Oncon0 6361  (class class class)co 7405  2oc2o 8456  o coe 8461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-oadd 8466  df-omul 8467  df-oexp 8468
This theorem is referenced by:  oeword  8586  infxpenc2lem1  10010
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