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Theorem rankprb 7733
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 7691 . . . 4  |-  ( A  e.  U. ( R1
" On )  ->  { A }  e.  U. ( R1 " On ) )
2 snwf 7691 . . . 4  |-  ( B  e.  U. ( R1
" On )  ->  { B }  e.  U. ( R1 " On ) )
3 rankunb 7732 . . . 4  |-  ( ( { A }  e.  U. ( R1 " On )  /\  { B }  e.  U. ( R1 " On ) )  ->  ( rank `  ( { A }  u.  { B } ) )  =  ( ( rank `  { A } )  u.  ( rank `  { B }
) ) )
41, 2, 3syl2an 464 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  ( { A }  u.  { B } ) )  =  ( ( rank `  { A } )  u.  ( rank `  { B }
) ) )
5 ranksnb 7709 . . . 4  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  { A } )  =  suc  ( rank `  A )
)
6 ranksnb 7709 . . . 4  |-  ( B  e.  U. ( R1
" On )  -> 
( rank `  { B } )  =  suc  ( rank `  B )
)
7 uneq12 3456 . . . 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 464 . . 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 2436 . 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 3781 . . 3  |-  { A ,  B }  =  ( { A }  u.  { B } )
1110fveq2i 5690 . 2  |-  ( rank `  { A ,  B } )  =  (
rank `  ( { A }  u.  { B } ) )
12 rankon 7677 . . . 4  |-  ( rank `  A )  e.  On
1312onordi 4645 . . 3  |-  Ord  ( rank `  A )
14 rankon 7677 . . . 4  |-  ( rank `  B )  e.  On
1514onordi 4645 . . 3  |-  Ord  ( rank `  B )
16 ordsucun 4764 . . 3  |-  ( ( Ord  ( rank `  A
)  /\  Ord  ( rank `  B ) )  ->  suc  ( ( rank `  A
)  u.  ( rank `  B ) )  =  ( suc  ( rank `  A )  u.  suc  ( rank `  B )
) )
1713, 15, 16mp2an 654 . 2  |-  suc  (
( rank `  A )  u.  ( rank `  B
) )  =  ( suc  ( rank `  A
)  u.  suc  ( rank `  B ) )
189, 11, 173eqtr4g 2461 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 359    = wceq 1649    e. wcel 1721    u. cun 3278   {csn 3774   {cpr 3775   U.cuni 3975   Ord word 4540   Oncon0 4541   suc csuc 4543   "cima 4840   ` cfv 5413   R1cr1 7644   rankcrnk 7645
This theorem is referenced by:  rankopb  7734  rankpr  7739  r1limwun  8567  rankaltopb  25728
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-recs 6592  df-rdg 6627  df-r1 7646  df-rank 7647
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