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Theorem ordgt0ge1 7522
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 5737 . . 3 ∅ ∈ On
2 ordelsuc 6967 . . 3 ((∅ ∈ On ∧ Ord 𝐴) → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴))
31, 2mpan 705 . 2 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴))
4 df-1o 7505 . . 3 1𝑜 = suc ∅
54sseq1i 3608 . 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 1987  wss 3555  c0 3891  Ord word 5681  Oncon0 5682  suc csuc 5684  1𝑜c1o 7498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-ord 5685  df-on 5686  df-suc 5688  df-1o 7505
This theorem is referenced by:  ordge1n0  7523  oe0m1  7546  omword1  7598  omword2  7599  omlimcl  7603  oen0  7611  oewordi  7616
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