Theorem ordgt0ge1 6681

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 4518	. . 3 |- (/) e. On
2		ordelsuc 4632	. . 3 |- ( ( (/) e. On /\ Ord A ) -> ( (/) e. A <-> suc (/) C_ A ) )
3	1, 2	mpan 424	. 2 |- ( Ord A -> ( (/) e. A <-> suc (/) C_ A ) )
4		df-1o 6660	. . 3 |- 1o = suc (/)
5	4	sseq1i 3268	. 2 |- ( 1o C_ A <-> suc (/) C_ A )
6	3, 5	bitr4di 198	1 |- ( Ord A -> ( (/) e. A <-> 1o C_ A ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2205   C_ wss 3214   (/)c0 3512   Ord word 4488   Oncon0 4489   suc csuc 4491   1oc1o 6653

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216  ax-nul 4241

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-uni 3920  df-tr 4214  df-iord 4492  df-on 4494  df-suc 4497  df-1o 6660

This theorem is referenced by:  ordge1n0im  6682  archnqq  7748