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Theorem ranksnb 7495
Description: The rank of a singleton. Theorem 15.17(v) of [Monk1] p. 112. (Contributed by Mario Carneiro, 10-Jun-2013.)
Assertion
Ref Expression
ranksnb  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  { A } )  =  suc  ( rank `  A )
)

Proof of Theorem ranksnb
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5486 . . . . . 6  |-  ( y  =  A  ->  ( rank `  y )  =  ( rank `  A
) )
21eleq1d 2350 . . . . 5  |-  ( y  =  A  ->  (
( rank `  y )  e.  x  <->  ( rank `  A
)  e.  x ) )
32ralsng 3673 . . . 4  |-  ( A  e.  U. ( R1
" On )  -> 
( A. y  e. 
{ A }  ( rank `  y )  e.  x  <->  ( rank `  A
)  e.  x ) )
43rabbidv 2781 . . 3  |-  ( A  e.  U. ( R1
" On )  ->  { x  e.  On  |  A. y  e.  { A }  ( rank `  y )  e.  x }  =  { x  e.  On  |  ( rank `  A )  e.  x } )
54inteqd 3868 . 2  |-  ( A  e.  U. ( R1
" On )  ->  |^| { x  e.  On  |  A. y  e.  { A }  ( rank `  y )  e.  x }  =  |^| { x  e.  On  |  ( rank `  A )  e.  x } )
6 snwf 7477 . . 3  |-  ( A  e.  U. ( R1
" On )  ->  { A }  e.  U. ( R1 " On ) )
7 rankval3b 7494 . . 3  |-  ( { A }  e.  U. ( R1 " On )  ->  ( rank `  { A } )  =  |^| { x  e.  On  |  A. y  e.  { A }  ( rank `  y
)  e.  x }
)
86, 7syl 15 . 2  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  { A } )  =  |^| { x  e.  On  |  A. y  e.  { A }  ( rank `  y
)  e.  x }
)
9 rankon 7463 . . 3  |-  ( rank `  A )  e.  On
10 onsucmin 4611 . . 3  |-  ( (
rank `  A )  e.  On  ->  suc  ( rank `  A )  =  |^| { x  e.  On  | 
( rank `  A )  e.  x } )
119, 10mp1i 11 . 2  |-  ( A  e.  U. ( R1
" On )  ->  suc  ( rank `  A
)  =  |^| { x  e.  On  |  ( rank `  A )  e.  x } )
125, 8, 113eqtr4d 2326 1  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  { A } )  =  suc  ( rank `  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1685   A.wral 2544   {crab 2548   {csn 3641   U.cuni 3828   |^|cint 3863   Oncon0 4391   suc csuc 4393   "cima 4691   ` cfv 5221   R1cr1 7430   rankcrnk 7431
This theorem is referenced by:  rankprb  7519  ranksn  7522  rankcf  8395  rankaltopb  23923
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-recs 6384  df-rdg 6419  df-r1 7432  df-rank 7433
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