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Theorem rankaltopb 29556
Description: Compute the rank of an alternate ordered pair. (Contributed by Scott Fenton, 18-Dec-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
rankaltopb  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  << A ,  B >> )  =  suc  suc  ( ( rank `  A
)  u.  suc  ( rank `  B ) ) )

Proof of Theorem rankaltopb
StepHypRef Expression
1 snwf 8239 . . 3  |-  ( B  e.  U. ( R1
" On )  ->  { B }  e.  U. ( R1 " On ) )
2 df-altop 29535 . . . . . 6  |-  << A ,  B >>  =  { { A } ,  { A ,  { B } } }
32fveq2i 5875 . . . . 5  |-  ( rank `  << A ,  B >> )  =  ( rank `  { { A } ,  { A ,  { B } } } )
4 snwf 8239 . . . . . . 7  |-  ( A  e.  U. ( R1
" On )  ->  { A }  e.  U. ( R1 " On ) )
54adantr 465 . . . . . 6  |-  ( ( A  e.  U. ( R1 " On )  /\  { B }  e.  U. ( R1 " On ) )  ->  { A }  e.  U. ( R1 " On ) )
6 prwf 8241 . . . . . 6  |-  ( ( A  e.  U. ( R1 " On )  /\  { B }  e.  U. ( R1 " On ) )  ->  { A ,  { B } }  e.  U. ( R1 " On ) )
7 rankprb 8281 . . . . . 6  |-  ( ( { A }  e.  U. ( R1 " On )  /\  { A ,  { B } }  e.  U. ( R1 " On ) )  ->  ( rank `  { { A } ,  { A ,  { B } } } )  =  suc  ( ( rank `  { A } )  u.  ( rank `  { A ,  { B } } ) ) )
85, 6, 7syl2anc 661 . . . . 5  |-  ( ( A  e.  U. ( R1 " On )  /\  { B }  e.  U. ( R1 " On ) )  ->  ( rank `  { { A } ,  { A ,  { B } } } )  =  suc  ( (
rank `  { A } )  u.  ( rank `  { A ,  { B } } ) ) )
93, 8syl5eq 2520 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  { B }  e.  U. ( R1 " On ) )  ->  ( rank ` 
<< A ,  B >> )  =  suc  ( (
rank `  { A } )  u.  ( rank `  { A ,  { B } } ) ) )
10 snsspr1 4182 . . . . . . . 8  |-  { A }  C_  { A ,  { B } }
11 ssequn1 3679 . . . . . . . 8  |-  ( { A }  C_  { A ,  { B } }  <->  ( { A }  u.  { A ,  { B } } )  =  { A ,  { B } } )
1210, 11mpbi 208 . . . . . . 7  |-  ( { A }  u.  { A ,  { B } } )  =  { A ,  { B } }
1312fveq2i 5875 . . . . . 6  |-  ( rank `  ( { A }  u.  { A ,  { B } } ) )  =  ( rank `  { A ,  { B } } )
14 rankunb 8280 . . . . . . 7  |-  ( ( { A }  e.  U. ( R1 " On )  /\  { A ,  { B } }  e.  U. ( R1 " On ) )  ->  ( rank `  ( { A }  u.  { A ,  { B } }
) )  =  ( ( rank `  { A } )  u.  ( rank `  { A ,  { B } } ) ) )
155, 6, 14syl2anc 661 . . . . . 6  |-  ( ( A  e.  U. ( R1 " On )  /\  { B }  e.  U. ( R1 " On ) )  ->  ( rank `  ( { A }  u.  { A ,  { B } } ) )  =  ( ( rank `  { A } )  u.  ( rank `  { A ,  { B } } ) ) )
16 rankprb 8281 . . . . . 6  |-  ( ( A  e.  U. ( R1 " On )  /\  { B }  e.  U. ( R1 " On ) )  ->  ( rank `  { A ,  { B } } )  =  suc  ( ( rank `  A )  u.  ( rank `  { B }
) ) )
1713, 15, 163eqtr3a 2532 . . . . 5  |-  ( ( A  e.  U. ( R1 " On )  /\  { B }  e.  U. ( R1 " On ) )  ->  ( ( rank `  { A }
)  u.  ( rank `  { A ,  { B } } ) )  =  suc  ( (
rank `  A )  u.  ( rank `  { B } ) ) )
18 suceq 4949 . . . . 5  |-  ( ( ( rank `  { A } )  u.  ( rank `  { A ,  { B } } ) )  =  suc  (
( rank `  A )  u.  ( rank `  { B } ) )  ->  suc  ( ( rank `  { A } )  u.  ( rank `  { A ,  { B } } ) )  =  suc  suc  ( ( rank `  A
)  u.  ( rank `  { B } ) ) )
1917, 18syl 16 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  { B }  e.  U. ( R1 " On ) )  ->  suc  ( (
rank `  { A } )  u.  ( rank `  { A ,  { B } } ) )  =  suc  suc  ( ( rank `  A
)  u.  ( rank `  { B } ) ) )
209, 19eqtrd 2508 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  { B }  e.  U. ( R1 " On ) )  ->  ( rank ` 
<< A ,  B >> )  =  suc  suc  (
( rank `  A )  u.  ( rank `  { B } ) ) )
211, 20sylan2 474 . 2  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  << A ,  B >> )  =  suc  suc  ( ( rank `  A
)  u.  ( rank `  { B } ) ) )
22 ranksnb 8257 . . . . 5  |-  ( B  e.  U. ( R1
" On )  -> 
( rank `  { B } )  =  suc  ( rank `  B )
)
2322uneq2d 3663 . . . 4  |-  ( B  e.  U. ( R1
" On )  -> 
( ( rank `  A
)  u.  ( rank `  { B } ) )  =  ( (
rank `  A )  u.  suc  ( rank `  B
) ) )
24 suceq 4949 . . . 4  |-  ( ( ( rank `  A
)  u.  ( rank `  { B } ) )  =  ( (
rank `  A )  u.  suc  ( rank `  B
) )  ->  suc  ( ( rank `  A
)  u.  ( rank `  { B } ) )  =  suc  (
( rank `  A )  u.  suc  ( rank `  B
) ) )
25 suceq 4949 . . . 4  |-  ( suc  ( ( rank `  A
)  u.  ( rank `  { B } ) )  =  suc  (
( rank `  A )  u.  suc  ( rank `  B
) )  ->  suc  suc  ( ( rank `  A
)  u.  ( rank `  { B } ) )  =  suc  suc  ( ( rank `  A
)  u.  suc  ( rank `  B ) ) )
2623, 24, 253syl 20 . . 3  |-  ( B  e.  U. ( R1
" On )  ->  suc  suc  ( ( rank `  A )  u.  ( rank `  { B }
) )  =  suc  suc  ( ( rank `  A
)  u.  suc  ( rank `  B ) ) )
2726adantl 466 . 2  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  suc  suc  ( (
rank `  A )  u.  ( rank `  { B } ) )  =  suc  suc  ( ( rank `  A )  u. 
suc  ( rank `  B
) ) )
2821, 27eqtrd 2508 1  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  << A ,  B >> )  =  suc  suc  ( ( rank `  A
)  u.  suc  ( rank `  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    u. cun 3479    C_ wss 3481   {csn 4033   {cpr 4035   U.cuni 4251   Oncon0 4884   suc csuc 4886   "cima 5008   ` cfv 5594   R1cr1 8192   rankcrnk 8193   <<caltop 29533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6696  df-recs 7054  df-rdg 7088  df-r1 8194  df-rank 8195  df-altop 29535
This theorem is referenced by: (None)
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