Theorem ordge1n0im 6263

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 6262	. 2 |- ( Ord A -> ( (/) e. A <-> 1o C_ A ) )
2		ne0i 3316	. 2 |- ( (/) e. A -> A =/= (/) )
3	1, 2	syl6bir 163	1 |- ( Ord A -> ( 1o C_ A -> A =/= (/) ) )

Syntax hints:   -> wi 4   e. wcel 1448   =/= wne 2267   C_ wss 3021   (/)c0 3310   Ord word 4222   1oc1o 6236

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-nul 3994

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-uni 3684  df-tr 3967  df-iord 4226  df-on 4228  df-suc 4231  df-1o 6243

This theorem is referenced by: (None)