ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ordgt0ge1 Structured version   GIF version

Theorem ordgt0ge1 5931
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 5914 . . 3 1𝑜 = suc ∅
54sseq1i 2944 . 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 1374  wss 2892  c0 3199  Ord word 4046  Oncon0 4047  suc csuc 4049  1𝑜c1o 5907
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 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-nul 3855
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-v 2535  df-dif 2895  df-un 2897  df-in 2899  df-ss 2906  df-nul 3200  df-pw 3334  df-sn 3354  df-uni 3553  df-tr 3827  df-iord 4050  df-on 4052  df-suc 4055  df-1o 5914
This theorem is referenced by:  ordge1n0im  5932  archnqq  6272
  Copyright terms: Public domain W3C validator