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Theorem ordge1n0im 5899
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 5898 . 2 (Ord A → (∅ A ↔ 1𝑜A))
2 ne0i 3208 . 2 (∅ AA ≠ ∅)
31, 2syl6bir 153 1 (Ord A → (1𝑜AA ≠ ∅))
Colors of variables: wff set class
Syntax hints:  wi 4   wcel 1375  wne 2186  wss 2895  c0 3202  Ord word 4023  1𝑜c1o 5874
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 532  ax-in2 533  ax-io 617  ax-5 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004  ax-nul 3835
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1329  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2287  df-rex 2288  df-v 2535  df-dif 2898  df-un 2900  df-in 2902  df-ss 2909  df-nul 3203  df-pw 3313  df-sn 3333  df-uni 3533  df-tr 3807  df-iord 4027  df-on 4028  df-suc 4031  df-1o 5881
This theorem is referenced by: (None)
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