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Theorem oe0m1 8142
Description: Ordinal exponentiation with zero mantissa and nonzero exponent. Proposition 8.31(2) of [TakeutiZaring] p. 67 and its converse. (Contributed by NM, 5-Jan-2005.)
Assertion
Ref Expression
oe0m1 (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ (∅ ↑o 𝐴) = ∅))

Proof of Theorem oe0m1
StepHypRef Expression
1 eloni 6200 . . 3 (𝐴 ∈ On → Ord 𝐴)
2 ordgt0ge1 8118 . . 3 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o𝐴))
31, 2syl 17 . 2 (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 1o𝐴))
4 oe0m 8139 . . . 4 (𝐴 ∈ On → (∅ ↑o 𝐴) = (1o𝐴))
54eqeq1d 2828 . . 3 (𝐴 ∈ On → ((∅ ↑o 𝐴) = ∅ ↔ (1o𝐴) = ∅))
6 ssdif0 4327 . . 3 (1o𝐴 ↔ (1o𝐴) = ∅)
75, 6syl6rbbr 291 . 2 (𝐴 ∈ On → (1o𝐴 ↔ (∅ ↑o 𝐴) = ∅))
83, 7bitrd 280 1 (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ (∅ ↑o 𝐴) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1530  wcel 2107  cdif 3937  wss 3940  c0 4295  Ord word 6189  Oncon0 6190  (class class class)co 7150  1oc1o 8091  o coe 8097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-suc 6196  df-iota 6313  df-fun 6356  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oexp 8104
This theorem is referenced by:  oev2  8144  oesuclem  8146  oecl  8158  oewordri  8213  oelim2  8216  oeoa  8218  oeoe  8220  cantnf  9150
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