Theorem ordge1n0im 6437

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 6436	. 2 |- ( Ord A -> ( (/) e. A <-> 1o C_ A ) )
2		ne0i 3430	. 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 3130   (/)c0 3423   Ord word 4363   1oc1o 6410

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 4130

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 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-uni 3811  df-tr 4103  df-iord 4367  df-on 4369  df-suc 4372  df-1o 6417

This theorem is referenced by: (None)