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Theorem oe0m 8129
 Description: Ordinal exponentiation with zero mantissa. (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 6213 . . 3 ∅ ∈ On
2 oev 8125 . . 3 ((∅ ∈ On ∧ 𝐴 ∈ On) → (∅ ↑o 𝐴) = if(∅ = ∅, (1o𝐴), (rec((𝑥 ∈ V ↦ (𝑥 ·o ∅)), 1o)‘𝐴)))
31, 2mpan 689 . 2 (𝐴 ∈ On → (∅ ↑o 𝐴) = if(∅ = ∅, (1o𝐴), (rec((𝑥 ∈ V ↦ (𝑥 ·o ∅)), 1o)‘𝐴)))
4 eqid 2798 . . 3 ∅ = ∅
54iftruei 4432 . 2 if(∅ = ∅, (1o𝐴), (rec((𝑥 ∈ V ↦ (𝑥 ·o ∅)), 1o)‘𝐴)) = (1o𝐴)
63, 5eqtrdi 2849 1 (𝐴 ∈ On → (∅ ↑o 𝐴) = (1o𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∖ cdif 3878  ∅c0 4243  ifcif 4425   ↦ cmpt 5111  Oncon0 6160  ‘cfv 6325  (class class class)co 7136  reccrdg 8031  1oc1o 8081   ·o comu 8086   ↑o coe 8087 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296  ax-un 7444 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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-suc 6166  df-iota 6284  df-fun 6327  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-1o 8088  df-oexp 8094 This theorem is referenced by:  oe0m0  8131  oe0m1  8132  cantnflem2  9140
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