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Theorem ordgt0ge1 5957
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 4095 . . 3 On
2 ordelsuc 4197 . . 3 ((∅ On Ord A) → (∅ A ↔ suc ∅ ⊆ A))
31, 2mpan 400 . 2 (Ord A → (∅ A ↔ suc ∅ ⊆ A))
4 df-1o 5940 . . 3 1𝑜 = suc ∅
54sseq1i 2963 . 2 (1𝑜A ↔ suc ∅ ⊆ A)
63, 5syl6bbr 187 1 (Ord A → (∅ A ↔ 1𝑜A))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   wcel 1390  wss 2911  c0 3218  Ord word 4065  Oncon0 4066  suc csuc 4068  1𝑜c1o 5933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-nul 3874
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071  df-suc 4074  df-1o 5940
This theorem is referenced by:  ordge1n0im  5958  archnqq  6400
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