Theorem ordgt0ge1 6041

Description: Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.)

Assertion

Ref	Expression
ordgt0ge1	|- ( Ord A -> ( (/) e. A <-> 1o C_ A ) )

Proof of Theorem ordgt0ge1

Step	Hyp	Ref	Expression
1		0elon 4147	. . 3 |- (/) e. On
2		ordelsuc 4249	. . 3 |- ( ( (/) e. On /\ Ord A ) -> ( (/) e. A <-> suc (/) C_ A ) )
3	1, 2	mpan 414	. 2 |- ( Ord A -> ( (/) e. A <-> suc (/) C_ A ) )
4		df-1o 6024	. . 3 |- 1o = suc (/)
5	4	sseq1i 3023	. 2 |- ( 1o C_ A <-> suc (/) C_ A )
6	3, 5	syl6bbr 196	1 |- ( Ord A -> ( (/) e. A <-> 1o C_ A ) )

Syntax hints:   -> wi 4   <-> wb 103   e. wcel 1433   C_ wss 2973   (/)c0 3251   Ord word 4117   Oncon0 4118   suc csuc 4120   1oc1o 6017

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-nul 3904

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-uni 3602  df-tr 3876  df-iord 4121  df-on 4123  df-suc 4126  df-1o 6024

This theorem is referenced by:  ordge1n0im  6042  archnqq  6607