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Theorem ordgt0ge1 7748
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 5939 . . 3 ∅ ∈ On
2 ordelsuc 7186 . . 3 ((∅ ∈ On ∧ Ord 𝐴) → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴))
31, 2mpan 708 . 2 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴))
4 df-1o 7730 . . 3 1𝑜 = suc ∅
54sseq1i 3770 . 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 2139  wss 3715  c0 4058  Ord word 5883  Oncon0 5884  suc csuc 5886  1𝑜c1o 7723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-tr 4905  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-ord 5887  df-on 5888  df-suc 5890  df-1o 7730
This theorem is referenced by:  ordge1n0  7749  oe0m1  7772  omword1  7824  omword2  7825  omlimcl  7829  oen0  7837  oewordi  7842
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