MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oe0m1 Structured version   Visualization version   GIF version

Theorem oe0m1 7586
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 5721 . . 3 (𝐴 ∈ On → Ord 𝐴)
2 ordgt0ge1 7562 . . 3 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1𝑜𝐴))
31, 2syl 17 . 2 (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 1𝑜𝐴))
4 oe0m 7583 . . . 4 (𝐴 ∈ On → (∅ ↑𝑜 𝐴) = (1𝑜𝐴))
54eqeq1d 2622 . . 3 (𝐴 ∈ On → ((∅ ↑𝑜 𝐴) = ∅ ↔ (1𝑜𝐴) = ∅))
6 ssdif0 3933 . . 3 (1𝑜𝐴 ↔ (1𝑜𝐴) = ∅)
75, 6syl6rbbr 279 . 2 (𝐴 ∈ On → (1𝑜𝐴 ↔ (∅ ↑𝑜 𝐴) = ∅))
83, 7bitrd 268 1 (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ (∅ ↑𝑜 𝐴) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1481  wcel 1988  cdif 3564  wss 3567  c0 3907  Ord word 5710  Oncon0 5711  (class class class)co 6635  1𝑜c1o 7538  𝑜 coe 7544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-suc 5717  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oexp 7551
This theorem is referenced by:  oev2  7588  oesuclem  7590  oecl  7602  oewordri  7657  oelim2  7660  oeoa  7662  oeoe  7664  cantnf  8575
  Copyright terms: Public domain W3C validator