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Theorem rankprb 7477
Description: The rank of an unordered pair. Part of Exercise 30 of [Enderton] p. 207. (Contributed by Mario Carneiro, 10-Jun-2013.)
Assertion
Ref Expression
rankprb  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  { A ,  B }
)  =  suc  (
( rank `  A )  u.  ( rank `  B
) ) )

Proof of Theorem rankprb
StepHypRef Expression
1 snwf 7435 . . . 4  |-  ( A  e.  U. ( R1
" On )  ->  { A }  e.  U. ( R1 " On ) )
2 snwf 7435 . . . 4  |-  ( B  e.  U. ( R1
" On )  ->  { B }  e.  U. ( R1 " On ) )
3 rankunb 7476 . . . 4  |-  ( ( { A }  e.  U. ( R1 " On )  /\  { B }  e.  U. ( R1 " On ) )  ->  ( rank `  ( { A }  u.  { B } ) )  =  ( ( rank `  { A } )  u.  ( rank `  { B }
) ) )
41, 2, 3syl2an 465 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  ( { A }  u.  { B } ) )  =  ( ( rank `  { A } )  u.  ( rank `  { B }
) ) )
5 ranksnb 7453 . . . 4  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  { A } )  =  suc  ( rank `  A )
)
6 ranksnb 7453 . . . 4  |-  ( B  e.  U. ( R1
" On )  -> 
( rank `  { B } )  =  suc  ( rank `  B )
)
7 uneq12 3285 . . . 4  |-  ( ( ( rank `  { A } )  =  suc  ( rank `  A )  /\  ( rank `  { B } )  =  suc  ( rank `  B )
)  ->  ( ( rank `  { A }
)  u.  ( rank `  { B } ) )  =  ( suc  ( rank `  A
)  u.  suc  ( rank `  B ) ) )
85, 6, 7syl2an 465 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( ( rank `  { A } )  u.  ( rank `  { B } ) )  =  ( suc  ( rank `  A )  u.  suc  ( rank `  B )
) )
94, 8eqtrd 2288 . 2  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  ( { A }  u.  { B } ) )  =  ( suc  ( rank `  A )  u.  suc  ( rank `  B )
) )
10 df-pr 3607 . . 3  |-  { A ,  B }  =  ( { A }  u.  { B } )
1110fveq2i 5447 . 2  |-  ( rank `  { A ,  B } )  =  (
rank `  ( { A }  u.  { B } ) )
12 rankon 7421 . . . 4  |-  ( rank `  A )  e.  On
1312onordi 4455 . . 3  |-  Ord  ( rank `  A )
14 rankon 7421 . . . 4  |-  ( rank `  B )  e.  On
1514onordi 4455 . . 3  |-  Ord  ( rank `  B )
16 ordsucun 4574 . . 3  |-  ( ( Ord  ( rank `  A
)  /\  Ord  ( rank `  B ) )  ->  suc  ( ( rank `  A
)  u.  ( rank `  B ) )  =  ( suc  ( rank `  A )  u.  suc  ( rank `  B )
) )
1713, 15, 16mp2an 656 . 2  |-  suc  (
( rank `  A )  u.  ( rank `  B
) )  =  ( suc  ( rank `  A
)  u.  suc  ( rank `  B ) )
189, 11, 173eqtr4g 2313 1  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  { A ,  B }
)  =  suc  (
( rank `  A )  u.  ( rank `  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621    u. cun 3111   {csn 3600   {cpr 3601   U.cuni 3787   Ord word 4349   Oncon0 4350   suc csuc 4352   "cima 4650   ` cfv 4659   R1cr1 7388   rankcrnk 7389
This theorem is referenced by:  rankopb  7478  rankpr  7483  r1limwun  8312  rankaltopb  23874
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-recs 6342  df-rdg 6377  df-r1 7390  df-rank 7391
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