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Theorem ordge1n0im 6238
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 6237 . 2 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o𝐴))
2 ne0i 3308 . 2 (∅ ∈ 𝐴𝐴 ≠ ∅)
31, 2syl6bir 163 1 (Ord 𝐴 → (1o𝐴𝐴 ≠ ∅))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1445  wne 2262  wss 3013  c0 3302  Ord word 4213  1oc1o 6212
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-nul 3986
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-uni 3676  df-tr 3959  df-iord 4217  df-on 4219  df-suc 4222  df-1o 6219
This theorem is referenced by: (None)
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