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Theorem rankprb 7779
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 7737 . . . 4  |-  ( A  e.  U. ( R1
" On )  ->  { A }  e.  U. ( R1 " On ) )
2 snwf 7737 . . . 4  |-  ( B  e.  U. ( R1
" On )  ->  { B }  e.  U. ( R1 " On ) )
3 rankunb 7778 . . . 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 7755 . . . 4  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  { A } )  =  suc  ( rank `  A )
)
6 ranksnb 7755 . . . 4  |-  ( B  e.  U. ( R1
" On )  -> 
( rank `  { B } )  =  suc  ( rank `  B )
)
7 uneq12 3498 . . . 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 2470 . 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 3823 . . 3  |-  { A ,  B }  =  ( { A }  u.  { B } )
1110fveq2i 5733 . 2  |-  ( rank `  { A ,  B } )  =  (
rank `  ( { A }  u.  { B } ) )
12 rankon 7723 . . . 4  |-  ( rank `  A )  e.  On
1312onordi 4688 . . 3  |-  Ord  ( rank `  A )
14 rankon 7723 . . . 4  |-  ( rank `  B )  e.  On
1514onordi 4688 . . 3  |-  Ord  ( rank `  B )
16 ordsucun 4807 . . 3  |-  ( ( Ord  ( rank `  A
)  /\  Ord  ( rank `  B ) )  ->  suc  ( ( rank `  A
)  u.  ( rank `  B ) )  =  ( suc  ( rank `  A )  u.  suc  ( rank `  B )
) )
1713, 15, 16mp2an 655 . 2  |-  suc  (
( rank `  A )  u.  ( rank `  B
) )  =  ( suc  ( rank `  A
)  u.  suc  ( rank `  B ) )
189, 11, 173eqtr4g 2495 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 4    /\ wa 360    = wceq 1653    e. wcel 1726    u. cun 3320   {csn 3816   {cpr 3817   U.cuni 4017   Ord word 4582   Oncon0 4583   suc csuc 4585   "cima 4883   ` cfv 5456   R1cr1 7690   rankcrnk 7691
This theorem is referenced by:  rankopb  7780  rankpr  7785  r1limwun  8613  rankaltopb  25826
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
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 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-recs 6635  df-rdg 6670  df-r1 7692  df-rank 7693
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