Metamath Proof Explorer

Theorem rankkardu

Description: An upper bound on the rank of a kard cardinal. (Contributed by BTernaryTau, 4-Jul-2026)

Ref Expression
Assertion rankkardu 
|- ( rank ` ( kard ` A ) ) C_ suc ( rank ` A )

Proof

Step Hyp Ref Expression
1 kardval 
 |-  ( kard ` A ) = Scott { x | x ~~ A }
2 1 fveq2i 
 |-  ( rank ` ( kard ` A ) ) = ( rank ` Scott { x | x ~~ A } )
3 enrefg 
 |-  ( A e. _V -> A ~~ A )
4 breq1 
 |-  ( x = A -> ( x ~~ A <-> A ~~ A ) )
5 4 elabg 
 |-  ( A e. _V -> ( A e. { x | x ~~ A } <-> A ~~ A ) )
6 3 5 mpbird 
 |-  ( A e. _V -> A e. { x | x ~~ A } )
7 rankscottu 
 |-  ( A e. { x | x ~~ A } -> ( rank ` Scott { x | x ~~ A } ) C_ suc ( rank ` A ) )
8 6 7 syl 
 |-  ( A e. _V -> ( rank ` Scott { x | x ~~ A } ) C_ suc ( rank ` A ) )
9 2 8 eqsstrid 
 |-  ( A e. _V -> ( rank ` ( kard ` A ) ) C_ suc ( rank ` A ) )
10 kardeq0 
 |-  ( ( kard ` A ) = (/) <-> -. A e. _V )
11 fvex 
 |-  ( kard ` A ) e. _V
12 11 rankeq0 
 |-  ( ( kard ` A ) = (/) <-> ( rank ` ( kard ` A ) ) = (/) )
13 0ss 
 |-  (/) C_ suc ( rank ` A )
14 sseq1 
 |-  ( ( rank ` ( kard ` A ) ) = (/) -> ( ( rank ` ( kard ` A ) ) C_ suc ( rank ` A ) <-> (/) C_ suc ( rank ` A ) ) )
15 13 14 mpbiri 
 |-  ( ( rank ` ( kard ` A ) ) = (/) -> ( rank ` ( kard ` A ) ) C_ suc ( rank ` A ) )
16 12 15 sylbi 
 |-  ( ( kard ` A ) = (/) -> ( rank ` ( kard ` A ) ) C_ suc ( rank ` A ) )
17 10 16 sylbir 
 |-  ( -. A e. _V -> ( rank ` ( kard ` A ) ) C_ suc ( rank ` A ) )
18 9 17 pm2.61i 
 |-  ( rank ` ( kard ` A ) ) C_ suc ( rank ` A )