Theorem ordge1n0im 6522

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 6521	. 2 |- ( Ord A -> ( (/) e. A <-> 1o C_ A ) )
2		ne0i 3467	. 2 |- ( (/) e. A -> A =/= (/) )
3	1, 2	biimtrrdi 164	1 |- ( Ord A -> ( 1o C_ A -> A =/= (/) ) )

Syntax hints:   -> wi 4   e. wcel 2176   =/= wne 2376   C_ wss 3166   (/)c0 3460   Ord word 4409   1oc1o 6495

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-nul 4170

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-uni 3851  df-tr 4143  df-iord 4413  df-on 4415  df-suc 4418  df-1o 6502

This theorem is referenced by: (None)