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Theorem ordgt0ge1 8549
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 6449 . . 3 ∅ ∈ On
2 ordelsuc 7856 . . 3 ((∅ ∈ On ∧ Ord 𝐴) → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴))
31, 2mpan 689 . 2 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴))
4 df-1o 8522 . . 3 1o = suc ∅
54sseq1i 4037 . 2 (1o𝐴 ↔ suc ∅ ⊆ 𝐴)
63, 5bitr4di 289 1 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2108  wss 3976  c0 4352  Ord word 6394  Oncon0 6395  suc csuc 6397  1oc1o 8515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-ord 6398  df-on 6399  df-suc 6401  df-1o 8522
This theorem is referenced by:  ordge1n0  8550  oe0m1  8577  omword1  8629  omword2  8630  omlimcl  8634  oen0  8642  oewordi  8647  oe0rif  43247
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