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Theorem oeword 8620
Description: Weak ordering property of ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)
Assertion
Ref Expression
oeword ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 ↔ (𝐶o 𝐴) ⊆ (𝐶o 𝐵)))

Proof of Theorem oeword
StepHypRef Expression
1 oeord 8618 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 ↔ (𝐶o 𝐴) ∈ (𝐶o 𝐵)))
2 oecan 8619 . . . . 5 ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐶o 𝐴) = (𝐶o 𝐵) ↔ 𝐴 = 𝐵))
323coml 1124 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐶o 𝐴) = (𝐶o 𝐵) ↔ 𝐴 = 𝐵))
43bicomd 222 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 = 𝐵 ↔ (𝐶o 𝐴) = (𝐶o 𝐵)))
51, 4orbi12d 916 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐴𝐵𝐴 = 𝐵) ↔ ((𝐶o 𝐴) ∈ (𝐶o 𝐵) ∨ (𝐶o 𝐴) = (𝐶o 𝐵))))
6 onsseleq 6417 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
763adant3 1129 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
8 eldifi 4126 . . . 4 (𝐶 ∈ (On ∖ 2o) → 𝐶 ∈ On)
9 id 22 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ On ∧ 𝐵 ∈ On))
10 oecl 8567 . . . . . 6 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶o 𝐴) ∈ On)
11 oecl 8567 . . . . . 6 ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶o 𝐵) ∈ On)
1210, 11anim12dan 617 . . . . 5 ((𝐶 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐶o 𝐴) ∈ On ∧ (𝐶o 𝐵) ∈ On))
13 onsseleq 6417 . . . . 5 (((𝐶o 𝐴) ∈ On ∧ (𝐶o 𝐵) ∈ On) → ((𝐶o 𝐴) ⊆ (𝐶o 𝐵) ↔ ((𝐶o 𝐴) ∈ (𝐶o 𝐵) ∨ (𝐶o 𝐴) = (𝐶o 𝐵))))
1412, 13syl 17 . . . 4 ((𝐶 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐶o 𝐴) ⊆ (𝐶o 𝐵) ↔ ((𝐶o 𝐴) ∈ (𝐶o 𝐵) ∨ (𝐶o 𝐴) = (𝐶o 𝐵))))
158, 9, 14syl2anr 595 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐶o 𝐴) ⊆ (𝐶o 𝐵) ↔ ((𝐶o 𝐴) ∈ (𝐶o 𝐵) ∨ (𝐶o 𝐴) = (𝐶o 𝐵))))
16153impa 1107 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐶o 𝐴) ⊆ (𝐶o 𝐵) ↔ ((𝐶o 𝐴) ∈ (𝐶o 𝐵) ∨ (𝐶o 𝐴) = (𝐶o 𝐵))))
175, 7, 163bitr4d 310 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 ↔ (𝐶o 𝐴) ⊆ (𝐶o 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wo 845  w3a 1084   = wceq 1534  wcel 2099  cdif 3944  wss 3947  Oncon0 6376  (class class class)co 7424  2oc2o 8490  o coe 8495
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 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-2o 8497  df-oadd 8500  df-omul 8501  df-oexp 8502
This theorem is referenced by:  oewordi  8621  cantnftermord  42986  omabs2  42998
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