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Theorem ranksnb 7679
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 5661 . . . . . 6  |-  ( y  =  A  ->  ( rank `  y )  =  ( rank `  A
) )
21eleq1d 2446 . . . . 5  |-  ( y  =  A  ->  (
( rank `  y )  e.  x  <->  ( rank `  A
)  e.  x ) )
32ralsng 3782 . . . 4  |-  ( A  e.  U. ( R1
" On )  -> 
( A. y  e. 
{ A }  ( rank `  y )  e.  x  <->  ( rank `  A
)  e.  x ) )
43rabbidv 2884 . . 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 3990 . 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 7661 . . 3  |-  ( A  e.  U. ( R1
" On )  ->  { A }  e.  U. ( R1 " On ) )
7 rankval3b 7678 . . 3  |-  ( { A }  e.  U. ( R1 " On )  ->  ( rank `  { A } )  =  |^| { x  e.  On  |  A. y  e.  { A }  ( rank `  y
)  e.  x }
)
86, 7syl 16 . 2  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  { A } )  =  |^| { x  e.  On  |  A. y  e.  { A }  ( rank `  y
)  e.  x }
)
9 rankon 7647 . . 3  |-  ( rank `  A )  e.  On
10 onsucmin 4734 . . 3  |-  ( (
rank `  A )  e.  On  ->  suc  ( rank `  A )  =  |^| { x  e.  On  | 
( rank `  A )  e.  x } )
119, 10mp1i 12 . 2  |-  ( A  e.  U. ( R1
" On )  ->  suc  ( rank `  A
)  =  |^| { x  e.  On  |  ( rank `  A )  e.  x } )
125, 8, 113eqtr4d 2422 1  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  { A } )  =  suc  ( rank `  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   A.wral 2642   {crab 2646   {csn 3750   U.cuni 3950   |^|cint 3985   Oncon0 4515   suc csuc 4517   "cima 4814   ` cfv 5387   R1cr1 7614   rankcrnk 7615
This theorem is referenced by:  rankprb  7703  ranksn  7706  rankcf  8578  rankaltopb  25531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-recs 6562  df-rdg 6597  df-r1 7616  df-rank 7617
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