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Theorem ordgt0ge1 7924
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 6082 . . 3 ∅ ∈ On
2 ordelsuc 7351 . . 3 ((∅ ∈ On ∧ Ord 𝐴) → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴))
31, 2mpan 677 . 2 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴))
4 df-1o 7905 . . 3 1o = suc ∅
54sseq1i 3885 . 2 (1o𝐴 ↔ suc ∅ ⊆ 𝐴)
63, 5syl6bbr 281 1 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wcel 2050  wss 3829  c0 4178  Ord word 6028  Oncon0 6029  suc csuc 6031  1oc1o 7898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-tr 5031  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-ord 6032  df-on 6033  df-suc 6035  df-1o 7905
This theorem is referenced by:  ordge1n0  7925  oe0m1  7948  omword1  8000  omword2  8001  omlimcl  8005  oen0  8013  oewordi  8018
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