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Theorem ordgt0ge1 7622
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 5816 . . 3 ∅ ∈ On
2 ordelsuc 7062 . . 3 ((∅ ∈ On ∧ Ord 𝐴) → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴))
31, 2mpan 706 . 2 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴))
4 df-1o 7605 . . 3 1𝑜 = suc ∅
54sseq1i 3662 . 2 (1𝑜𝐴 ↔ suc ∅ ⊆ 𝐴)
63, 5syl6bbr 278 1 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1𝑜𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wcel 2030  wss 3607  c0 3948  Ord word 5760  Oncon0 5761  suc csuc 5763  1𝑜c1o 7598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-suc 5767  df-1o 7605
This theorem is referenced by:  ordge1n0  7623  oe0m1  7646  omword1  7698  omword2  7699  omlimcl  7703  oen0  7711  oewordi  7716
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