Metamath Proof Explorer


Theorem rankpr

Description: The rank of an unordered pair. Part of Exercise 30 of Enderton p. 207. (Contributed by NM, 28-Nov-2003) (Revised by Mario Carneiro, 17-Nov-2014)

Ref Expression
Hypotheses ranksn.1
|- A e. _V
rankun.2
|- B e. _V
Assertion rankpr
|- ( rank ` { A , B } ) = suc ( ( rank ` A ) u. ( rank ` B ) )

Proof

Step Hyp Ref Expression
1 ranksn.1
 |-  A e. _V
2 rankun.2
 |-  B e. _V
3 unir1
 |-  U. ( R1 " On ) = _V
4 1 3 eleqtrri
 |-  A e. U. ( R1 " On )
5 2 3 eleqtrri
 |-  B e. U. ( R1 " On )
6 rankprb
 |-  ( ( A e. U. ( R1 " On ) /\ B e. U. ( R1 " On ) ) -> ( rank ` { A , B } ) = suc ( ( rank ` A ) u. ( rank ` B ) ) )
7 4 5 6 mp2an
 |-  ( rank ` { A , B } ) = suc ( ( rank ` A ) u. ( rank ` B ) )