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Theorem ordgt0ge1 8443
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 6375 . . 3 ∅ ∈ On
2 ordelsuc 7759 . . 3 ((∅ ∈ On ∧ Ord 𝐴) → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴))
31, 2mpan 689 . 2 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴))
4 df-1o 8416 . . 3 1o = suc ∅
54sseq1i 3976 . 2 (1o𝐴 ↔ suc ∅ ⊆ 𝐴)
63, 5bitr4di 289 1 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2107  wss 3914  c0 4286  Ord word 6320  Oncon0 6321  suc csuc 6323  1oc1o 8409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-tr 5227  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-ord 6324  df-on 6325  df-suc 6327  df-1o 8416
This theorem is referenced by:  ordge1n0  8444  oe0m1  8471  omword1  8524  omword2  8525  omlimcl  8529  oen0  8537  oewordi  8542  oe0rif  41667
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