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Theorem ordge1n0im 6132
Description: An ordinal greater than or equal to 1 is nonzero. (Contributed by Jim Kingdon, 26-Jun-2019.)
Assertion
Ref Expression
ordge1n0im (Ord 𝐴 → (1𝑜𝐴𝐴 ≠ ∅))

Proof of Theorem ordge1n0im
StepHypRef Expression
1 ordgt0ge1 6131 . 2 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1𝑜𝐴))
2 ne0i 3275 . 2 (∅ ∈ 𝐴𝐴 ≠ ∅)
31, 2syl6bir 162 1 (Ord 𝐴 → (1𝑜𝐴𝐴 ≠ ∅))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1434  wne 2249  wss 2984  c0 3269  Ord word 4153  1𝑜c1o 6106
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-nul 3930
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-uni 3628  df-tr 3902  df-iord 4157  df-on 4159  df-suc 4162  df-1o 6113
This theorem is referenced by: (None)
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