Theorem ordgt0ge1 6698

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 4532	. . 3 |- (/) e. On
2		ordelsuc 4647	. . 3 |- ( ( (/) e. On /\ Ord A ) -> ( (/) e. A <-> suc (/) C_ A ) )
3	1, 2	mpan 428	. 2 |- ( Ord A -> ( (/) e. A <-> suc (/) C_ A ) )
4		df-1o 6677	. . 3 |- 1o = suc (/)
5	4	sseq1i 3274	. 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 2209   C_ wss 3220   (/)c0 3520   Ord word 4502   Oncon0 4503   suc csuc 4505   1oc1o 6670

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-nul 4254

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-uni 3931  df-tr 4225  df-iord 4506  df-on 4508  df-suc 4511  df-1o 6677

This theorem is referenced by:  ordge1n0im  6699  archnqq  7774