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

Theorem ordgt0ge1 6300
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 4284 . . 3 ∅ ∈ On
2 ordelsuc 4391 . . 3 ((∅ ∈ On ∧ Ord 𝐴) → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴))
31, 2mpan 420 . 2 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴))
4 df-1o 6281 . . 3 1o = suc ∅
54sseq1i 3093 . 2 (1o𝐴 ↔ suc ∅ ⊆ 𝐴)
63, 5syl6bbr 197 1 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wcel 1465  wss 3041  c0 3333  Ord word 4254  Oncon0 4255  suc csuc 4257  1oc1o 6274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-nul 4024
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-uni 3707  df-tr 3997  df-iord 4258  df-on 4260  df-suc 4263  df-1o 6281
This theorem is referenced by:  ordge1n0im  6301  archnqq  7193
  Copyright terms: Public domain W3C validator