MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ranksnb Structured version   Unicode version

Theorem ranksnb 7756
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 5731 . . . . . 6  |-  ( y  =  A  ->  ( rank `  y )  =  ( rank `  A
) )
21eleq1d 2504 . . . . 5  |-  ( y  =  A  ->  (
( rank `  y )  e.  x  <->  ( rank `  A
)  e.  x ) )
32ralsng 3848 . . . 4  |-  ( A  e.  U. ( R1
" On )  -> 
( A. y  e. 
{ A }  ( rank `  y )  e.  x  <->  ( rank `  A
)  e.  x ) )
43rabbidv 2950 . . 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 4057 . 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 7738 . . 3  |-  ( A  e.  U. ( R1
" On )  ->  { A }  e.  U. ( R1 " On ) )
7 rankval3b 7755 . . 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 7724 . . 3  |-  ( rank `  A )  e.  On
10 onsucmin 4804 . . 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 2480 1  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  { A } )  =  suc  ( rank `  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   A.wral 2707   {crab 2711   {csn 3816   U.cuni 4017   |^|cint 4052   Oncon0 4584   suc csuc 4586   "cima 4884   ` cfv 5457   R1cr1 7691   rankcrnk 7692
This theorem is referenced by:  rankprb  7780  ranksn  7783  rankcf  8657  rankaltopb  25829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-recs 6636  df-rdg 6671  df-r1 7693  df-rank 7694
  Copyright terms: Public domain W3C validator