This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem ssrankr1

Description: A relationship between an ordinal number less than or equal to a rank, and the cumulative hierarchy of sets R1 . Proposition 9.15(3) of TakeutiZaring p. 79. (Contributed by NM, 8-Oct-2003) (Revised by Mario Carneiro, 17-Nov-2014)

Ref Expression
Hypothesis rankid.1 
|- A e. _V
Assertion ssrankr1 
|- ( B e. On -> ( B C_ ( rank ` A ) <-> -. A e. ( R1 ` B ) ) )

Proof

Step Hyp Ref Expression
1 rankid.1 
 |-  A e. _V
2 unir1 
 |-  U. ( R1 " On ) = _V
3 1 2 eleqtrri 
 |-  A e. U. ( R1 " On )
4 r1fnon 
 |-  R1 Fn On
5 fndm 
 |-  ( R1 Fn On -> dom R1 = On )
6 4 5 ax-mp 
 |-  dom R1 = On
7 6 eleq2i 
 |-  ( B e. dom R1 <-> B e. On )
8 7 biimpri 
 |-  ( B e. On -> B e. dom R1 )
9 rankr1clem 
 |-  ( ( A e. U. ( R1 " On ) /\ B e. dom R1 ) -> ( -. A e. ( R1 ` B ) <-> B C_ ( rank ` A ) ) )
10 3 8 9 sylancr 
 |-  ( B e. On -> ( -. A e. ( R1 ` B ) <-> B C_ ( rank ` A ) ) )
11 10 bicomd 
 |-  ( B e. On -> ( B C_ ( rank ` A ) <-> -. A e. ( R1 ` B ) ) )