Theorem ordge1n0im 6434

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

Step	Hyp	Ref	Expression
1		ordgt0ge1 6433	. 2 |- ( Ord A -> ( (/) e. A <-> 1o C_ A ) )
2		ne0i 3429	. 2 |- ( (/) e. A -> A =/= (/) )
3	1, 2	syl6bir 164	1 |- ( Ord A -> ( 1o C_ A -> A =/= (/) ) )

Syntax hints:   -> wi 4   e. wcel 2148   =/= wne 2347   C_ wss 3129   (/)c0 3422   Ord word 4361   1oc1o 6407

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-nul 4128

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-uni 3810  df-tr 4101  df-iord 4365  df-on 4367  df-suc 4370  df-1o 6414

This theorem is referenced by: (None)