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Theorem ordge1n0im 6077
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 6076 . 2 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1𝑜𝐴))
2 ne0i 3258 . 2 (∅ ∈ 𝐴𝐴 ≠ ∅)
31, 2syl6bir 162 1 (Ord 𝐴 → (1𝑜𝐴𝐴 ≠ ∅))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1434  wne 2246  wss 2974  c0 3252  Ord word 4119  1𝑜c1o 6052
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 2064  ax-nul 3906
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-pw 3386  df-sn 3406  df-uni 3604  df-tr 3878  df-iord 4123  df-on 4125  df-suc 4128  df-1o 6059
This theorem is referenced by: (None)
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