Theorem ordge1n0im 6494

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 6493	. 2 |- ( Ord A -> ( (/) e. A <-> 1o C_ A ) )
2		ne0i 3457	. 2 |- ( (/) e. A -> A =/= (/) )
3	1, 2	biimtrrdi 164	1 |- ( Ord A -> ( 1o C_ A -> A =/= (/) ) )

Syntax hints:   -> wi 4   e. wcel 2167   =/= wne 2367   C_ wss 3157   (/)c0 3450   Ord word 4397   1oc1o 6467

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-nul 4159

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-uni 3840  df-tr 4132  df-iord 4401  df-on 4403  df-suc 4406  df-1o 6474

This theorem is referenced by: (None)