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Theorem ordge1n0im 6398
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 6397 . 2 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o𝐴))
2 ne0i 3413 . 2 (∅ ∈ 𝐴𝐴 ≠ ∅)
31, 2syl6bir 163 1 (Ord 𝐴 → (1o𝐴𝐴 ≠ ∅))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2135  wne 2334  wss 3114  c0 3407  Ord word 4337  1oc1o 6371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146  ax-nul 4105
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-v 2726  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3408  df-pw 3558  df-sn 3579  df-uni 3787  df-tr 4078  df-iord 4341  df-on 4343  df-suc 4346  df-1o 6378
This theorem is referenced by: (None)
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