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Theorem ordge1n0im 5982
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 5981 . 2 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1𝑜𝐴))
2 ne0i 3227 . 2 (∅ ∈ 𝐴𝐴 ≠ ∅)
31, 2syl6bir 153 1 (Ord 𝐴 → (1𝑜𝐴𝐴 ≠ ∅))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1393  wne 2204  wss 2914  c0 3221  Ord word 4071  1𝑜c1o 5957
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 3880
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-ne 2206  df-ral 2308  df-rex 2309  df-v 2556  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-uni 3578  df-tr 3852  df-iord 4075  df-on 4077  df-suc 4080  df-1o 5964
This theorem is referenced by: (None)
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