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Theorem rankprb 7456
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 7414 . . . 4  |-  ( A  e.  U. ( R1
" On )  ->  { A }  e.  U. ( R1 " On ) )
2 snwf 7414 . . . 4  |-  ( B  e.  U. ( R1
" On )  ->  { B }  e.  U. ( R1 " On ) )
3 rankunb 7455 . . . 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 7432 . . . 4  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  { A } )  =  suc  ( rank `  A )
)
6 ranksnb 7432 . . . 4  |-  ( B  e.  U. ( R1
" On )  -> 
( rank `  { B } )  =  suc  ( rank `  B )
)
7 uneq12 3266 . . . 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 3588 . . 3  |-  { A ,  B }  =  ( { A }  u.  { B } )
1110fveq2i 5426 . 2  |-  ( rank `  { A ,  B } )  =  (
rank `  ( { A }  u.  { B } ) )
12 rankon 7400 . . . 4  |-  ( rank `  A )  e.  On
1312onordi 4434 . . 3  |-  Ord  ( rank `  A )
14 rankon 7400 . . . 4  |-  ( rank `  B )  e.  On
1514onordi 4434 . . 3  |-  Ord  ( rank `  B )
16 ordsucun 4553 . . 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 3092   {csn 3581   {cpr 3582   U.cuni 3768   Ord word 4328   Oncon0 4329   suc csuc 4331   "cima 4629   ` cfv 4638   R1cr1 7367   rankcrnk 7368
This theorem is referenced by:  rankopb  7457  rankpr  7462  r1limwun  8291  rankaltopb  23853
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 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
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 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-recs 6321  df-rdg 6356  df-r1 7369  df-rank 7370
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