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Theorem ordgt0ge1 5928
Description: Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.)
Assertion
Ref Expression
ordgt0ge1 (Ord A → (∅ A ↔ 1𝑜A))

Proof of Theorem ordgt0ge1
StepHypRef Expression
1 0elon 4076 . . 3 On
2 ordelsuc 4179 . . 3 ((∅ On Ord A) → (∅ A ↔ suc ∅ ⊆ A))
31, 2mpan 402 . 2 (Ord A → (∅ A ↔ suc ∅ ⊆ A))
4 df-1o 5911 . . 3 1𝑜 = suc ∅
54sseq1i 2945 . 2 (1𝑜A ↔ suc ∅ ⊆ A)
63, 5syl6bbr 187 1 (Ord A → (∅ A ↔ 1𝑜A))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   wcel 1375  wss 2893  c0 3200  Ord word 4046  Oncon0 4047  suc csuc 4049  1𝑜c1o 5904
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005  ax-nul 3856
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-rex 2289  df-v 2536  df-dif 2896  df-un 2898  df-in 2900  df-ss 2907  df-nul 3201  df-pw 3335  df-sn 3355  df-uni 3554  df-tr 3828  df-iord 4050  df-on 4052  df-suc 4055  df-1o 5911
This theorem is referenced by:  ordge1n0im  5929  archnqq  6261
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