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Theorem ordgt0ge1 8522
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 6428 . . 3 ∅ ∈ On
2 ordelsuc 7831 . . 3 ((∅ ∈ On ∧ Ord 𝐴) → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴))
31, 2mpan 688 . 2 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴))
4 df-1o 8495 . . 3 1o = suc ∅
54sseq1i 4010 . 2 (1o𝐴 ↔ suc ∅ ⊆ 𝐴)
63, 5bitr4di 288 1 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2098  wss 3949  c0 4326  Ord word 6373  Oncon0 6374  suc csuc 6376  1oc1o 8488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-tr 5270  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-ord 6377  df-on 6378  df-suc 6380  df-1o 8495
This theorem is referenced by:  ordge1n0  8523  oe0m1  8550  omword1  8602  omword2  8603  omlimcl  8607  oen0  8615  oewordi  8620  oe0rif  42763
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