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Theorem ordge1n0im 6156
Description: An ordinal greater than or equal to 1 is nonzero. (Contributed by Jim Kingdon, 26-Jun-2019.)
Assertion
Ref Expression
ordge1n0im  |-  ( Ord 
A  ->  ( 1o  C_  A  ->  A  =/=  (/) ) )

Proof of Theorem ordge1n0im
StepHypRef Expression
1 ordgt0ge1 6155 . 2  |-  ( Ord 
A  ->  ( (/)  e.  A  <->  1o  C_  A ) )
2 ne0i 3281 . 2  |-  ( (/)  e.  A  ->  A  =/=  (/) )
31, 2syl6bir 162 1  |-  ( Ord 
A  ->  ( 1o  C_  A  ->  A  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1436    =/= wne 2251    C_ wss 2988   (/)c0 3275   Ord word 4165   1oc1o 6130
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-nul 3942
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-v 2617  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-uni 3639  df-tr 3914  df-iord 4169  df-on 4171  df-suc 4174  df-1o 6137
This theorem is referenced by: (None)
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