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Theorem ordgt0ge1 6213
 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 4228 . . 3 ∅ ∈ On
2 ordelsuc 4335 . . 3 ((∅ ∈ On ∧ Ord 𝐴) → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴))
31, 2mpan 416 . 2 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴))
4 df-1o 6195 . . 3 1o = suc ∅
54sseq1i 3051 . 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 1439   ⊆ wss 3000  ∅c0 3287  Ord word 4198  Oncon0 4199  suc csuc 4201  1oc1o 6188 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-nul 3971 This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-uni 3660  df-tr 3943  df-iord 4202  df-on 4204  df-suc 4207  df-1o 6195 This theorem is referenced by:  ordge1n0im  6214  archnqq  7037
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