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Theorem oe0m 8245
Description: Value of zero raised to an ordinal. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
oe0m (𝐴 ∈ On → (∅ ↑o 𝐴) = (1o𝐴))

Proof of Theorem oe0m
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 0elon 6266 . . 3 ∅ ∈ On
2 oev 8241 . . 3 ((∅ ∈ On ∧ 𝐴 ∈ On) → (∅ ↑o 𝐴) = if(∅ = ∅, (1o𝐴), (rec((𝑥 ∈ V ↦ (𝑥 ·o ∅)), 1o)‘𝐴)))
31, 2mpan 690 . 2 (𝐴 ∈ On → (∅ ↑o 𝐴) = if(∅ = ∅, (1o𝐴), (rec((𝑥 ∈ V ↦ (𝑥 ·o ∅)), 1o)‘𝐴)))
4 eqid 2737 . . 3 ∅ = ∅
54iftruei 4446 . 2 if(∅ = ∅, (1o𝐴), (rec((𝑥 ∈ V ↦ (𝑥 ·o ∅)), 1o)‘𝐴)) = (1o𝐴)
63, 5eqtrdi 2794 1 (𝐴 ∈ On → (∅ ↑o 𝐴) = (1o𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2110  Vcvv 3408  cdif 3863  c0 4237  ifcif 4439  cmpt 5135  Oncon0 6213  cfv 6380  (class class class)co 7213  reccrdg 8145  1oc1o 8195   ·o comu 8200  o coe 8201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-suc 6219  df-iota 6338  df-fun 6382  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-oexp 8208
This theorem is referenced by:  oe0m0  8247  oe0m1  8248  cantnflem2  9305
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