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Theorem oe0 7870
 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 6913 . . . . 5 (𝐴 = ∅ → (𝐴o ∅) = (∅ ↑o ∅))
2 oe0m0 7868 . . . . 5 (∅ ↑o ∅) = 1o
31, 2syl6eq 2878 . . . 4 (𝐴 = ∅ → (𝐴o ∅) = 1o)
43adantl 475 . . 3 ((𝐴 ∈ On ∧ 𝐴 = ∅) → (𝐴o ∅) = 1o)
5 0elon 6017 . . . . . 6 ∅ ∈ On
6 oevn0 7863 . . . . . 6 (((𝐴 ∈ On ∧ ∅ ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴o ∅) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘∅))
75, 6mpanl2 694 . . . . 5 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴o ∅) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘∅))
8 1oex 7835 . . . . . 6 1o ∈ V
98rdg0 7784 . . . . 5 (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘∅) = 1o
107, 9syl6eq 2878 . . . 4 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴o ∅) = 1o)
1110adantll 707 . . 3 (((𝐴 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴o ∅) = 1o)
124, 11oe0lem 7861 . 2 ((𝐴 ∈ On ∧ 𝐴 ∈ On) → (𝐴o ∅) = 1o)
1312anidms 564 1 (𝐴 ∈ On → (𝐴o ∅) = 1o)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1658   ∈ wcel 2166  Vcvv 3415  ∅c0 4145   ↦ cmpt 4953  Oncon0 5964  ‘cfv 6124  (class class class)co 6906  reccrdg 7772  1oc1o 7820   ·o comu 7825   ↑o coe 7826 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-ord 5967  df-on 5968  df-lim 5969  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-om 7328  df-wrecs 7673  df-recs 7735  df-rdg 7773  df-1o 7827  df-oexp 7833 This theorem is referenced by:  oecl  7885  oe1  7892  oe1m  7893  oen0  7934  oewordri  7940  oeoalem  7944  oeoelem  7946  oeoe  7947  oeeulem  7949  nnecl  7961  oaabs2  7993  cantnff  8849
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