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Theorem oecan 7714
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 7712 . . . . . . 7 ((𝐶 ∈ On ∧ 𝐴 ∈ (On ∖ 2𝑜)) → (𝐵𝐶 → (𝐴𝑜 𝐵) ∈ (𝐴𝑜 𝐶)))
21ancoms 468 . . . . . 6 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐶 ∈ On) → (𝐵𝐶 → (𝐴𝑜 𝐵) ∈ (𝐴𝑜 𝐶)))
323adant2 1100 . . . . 5 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝐶 → (𝐴𝑜 𝐵) ∈ (𝐴𝑜 𝐶)))
4 oeordi 7712 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐴 ∈ (On ∖ 2𝑜)) → (𝐶𝐵 → (𝐴𝑜 𝐶) ∈ (𝐴𝑜 𝐵)))
54ancoms 468 . . . . . 6 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On) → (𝐶𝐵 → (𝐴𝑜 𝐶) ∈ (𝐴𝑜 𝐵)))
653adant3 1101 . . . . 5 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶𝐵 → (𝐴𝑜 𝐶) ∈ (𝐴𝑜 𝐵)))
73, 6orim12d 901 . . . 4 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵𝐶𝐶𝐵) → ((𝐴𝑜 𝐵) ∈ (𝐴𝑜 𝐶) ∨ (𝐴𝑜 𝐶) ∈ (𝐴𝑜 𝐵))))
87con3d 148 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ ((𝐴𝑜 𝐵) ∈ (𝐴𝑜 𝐶) ∨ (𝐴𝑜 𝐶) ∈ (𝐴𝑜 𝐵)) → ¬ (𝐵𝐶𝐶𝐵)))
9 eldifi 3765 . . . . . 6 (𝐴 ∈ (On ∖ 2𝑜) → 𝐴 ∈ On)
1093ad2ant1 1102 . . . . 5 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐴 ∈ On)
11 simp2 1082 . . . . 5 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐵 ∈ On)
12 oecl 7662 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) ∈ On)
1310, 11, 12syl2anc 694 . . . 4 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝑜 𝐵) ∈ On)
14 simp3 1083 . . . . 5 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐶 ∈ On)
15 oecl 7662 . . . . 5 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝑜 𝐶) ∈ On)
1610, 14, 15syl2anc 694 . . . 4 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝑜 𝐶) ∈ On)
17 eloni 5771 . . . . 5 ((𝐴𝑜 𝐵) ∈ On → Ord (𝐴𝑜 𝐵))
18 eloni 5771 . . . . 5 ((𝐴𝑜 𝐶) ∈ On → Ord (𝐴𝑜 𝐶))
19 ordtri3 5797 . . . . 5 ((Ord (𝐴𝑜 𝐵) ∧ Ord (𝐴𝑜 𝐶)) → ((𝐴𝑜 𝐵) = (𝐴𝑜 𝐶) ↔ ¬ ((𝐴𝑜 𝐵) ∈ (𝐴𝑜 𝐶) ∨ (𝐴𝑜 𝐶) ∈ (𝐴𝑜 𝐵))))
2017, 18, 19syl2an 493 . . . 4 (((𝐴𝑜 𝐵) ∈ On ∧ (𝐴𝑜 𝐶) ∈ On) → ((𝐴𝑜 𝐵) = (𝐴𝑜 𝐶) ↔ ¬ ((𝐴𝑜 𝐵) ∈ (𝐴𝑜 𝐶) ∨ (𝐴𝑜 𝐶) ∈ (𝐴𝑜 𝐵))))
2113, 16, 20syl2anc 694 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝑜 𝐵) = (𝐴𝑜 𝐶) ↔ ¬ ((𝐴𝑜 𝐵) ∈ (𝐴𝑜 𝐶) ∨ (𝐴𝑜 𝐶) ∈ (𝐴𝑜 𝐵))))
22 eloni 5771 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
23 eloni 5771 . . . . 5 (𝐶 ∈ On → Ord 𝐶)
24 ordtri3 5797 . . . . 5 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
2522, 23, 24syl2an 493 . . . 4 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
26253adant1 1099 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝐶𝐶𝐵)))
278, 21, 263imtr4d 283 . 2 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝑜 𝐵) = (𝐴𝑜 𝐶) → 𝐵 = 𝐶))
28 oveq2 6698 . 2 (𝐵 = 𝐶 → (𝐴𝑜 𝐵) = (𝐴𝑜 𝐶))
2927, 28impbid1 215 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 196  wo 382  w3a 1054   = wceq 1523  wcel 2030  cdif 3604  Ord word 5760  Oncon0 5761  (class class class)co 6690  2𝑜c2o 7599  𝑜 coe 7604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-omul 7610  df-oexp 7611
This theorem is referenced by:  oeword  7715  infxpenc2lem1  8880
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