Metamath Proof Explorer


Theorem 1sdom2ALT

Description: Alternate proof of 1sdom2 , shorter but requiring ax-un . (Contributed by NM, 4-Apr-2007) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion 1sdom2ALT
|- 1o ~< 2o

Proof

Step Hyp Ref Expression
1 1onn
 |-  1o e. _om
2 php4
 |-  ( 1o e. _om -> 1o ~< suc 1o )
3 1 2 ax-mp
 |-  1o ~< suc 1o
4 df-2o
 |-  2o = suc 1o
5 3 4 breqtrri
 |-  1o ~< 2o