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Theorem ordgt0ge1 6018
 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 4129 . . 3 ∅ ∈ On
2 ordelsuc 4231 . . 3 ((∅ ∈ On ∧ Ord 𝐴) → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴))
31, 2mpan 400 . 2 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴))
4 df-1o 6001 . . 3 1𝑜 = suc ∅
54sseq1i 2969 . 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 1393   ⊆ wss 2917  ∅c0 3224  Ord word 4099  Oncon0 4100  suc csuc 4102  1𝑜c1o 5994 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-nul 3883 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105  df-suc 4108  df-1o 6001 This theorem is referenced by:  ordge1n0im  6019  archnqq  6515
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