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Theorem oe0 8132
Description: Ordinal exponentiation with zero exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
oe0 (𝐴 ∈ On → (𝐴o ∅) = 1o)

Proof of Theorem oe0
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 oveq1 7147 . . . . 5 (𝐴 = ∅ → (𝐴o ∅) = (∅ ↑o ∅))
2 oe0m0 8130 . . . . 5 (∅ ↑o ∅) = 1o
31, 2syl6eq 2875 . . . 4 (𝐴 = ∅ → (𝐴o ∅) = 1o)
43adantl 485 . . 3 ((𝐴 ∈ On ∧ 𝐴 = ∅) → (𝐴o ∅) = 1o)
5 0elon 6227 . . . . . 6 ∅ ∈ On
6 oevn0 8125 . . . . . 6 (((𝐴 ∈ On ∧ ∅ ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴o ∅) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘∅))
75, 6mpanl2 700 . . . . 5 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴o ∅) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘∅))
8 1oex 8095 . . . . . 6 1o ∈ V
98rdg0 8042 . . . . 5 (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘∅) = 1o
107, 9syl6eq 2875 . . . 4 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴o ∅) = 1o)
1110adantll 713 . . 3 (((𝐴 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴o ∅) = 1o)
124, 11oe0lem 8123 . 2 ((𝐴 ∈ On ∧ 𝐴 ∈ On) → (𝐴o ∅) = 1o)
1312anidms 570 1 (𝐴 ∈ On → (𝐴o ∅) = 1o)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  Vcvv 3479  c0 4274  cmpt 5129  Oncon0 6174  cfv 6338  (class class class)co 7140  reccrdg 8030  1oc1o 8080   ·o comu 8085  o coe 8086
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4822  df-iun 4904  df-br 5050  df-opab 5112  df-mpt 5130  df-tr 5156  df-id 5443  df-eprel 5448  df-po 5457  df-so 5458  df-fr 5497  df-we 5499  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6297  df-fun 6340  df-fn 6341  df-f 6342  df-f1 6343  df-fo 6344  df-f1o 6345  df-fv 6346  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-1o 8087  df-oexp 8093
This theorem is referenced by:  oecl  8147  oe1  8155  oe1m  8156  oen0  8197  oewordri  8203  oeoalem  8207  oeoelem  8209  oeoe  8210  oeeulem  8212  nnecl  8224  oaabs2  8257  cantnff  9123
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