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Theorem rankaltopb 28174
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 8130 . . 3  |-  ( B  e.  U. ( R1
" On )  ->  { B }  e.  U. ( R1 " On ) )
2 df-altop 28153 . . . . . 6  |-  << A ,  B >>  =  { { A } ,  { A ,  { B } } }
32fveq2i 5805 . . . . 5  |-  ( rank `  << A ,  B >> )  =  ( rank `  { { A } ,  { A ,  { B } } } )
4 snwf 8130 . . . . . . 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 8132 . . . . . 6  |-  ( ( A  e.  U. ( R1 " On )  /\  { B }  e.  U. ( R1 " On ) )  ->  { A ,  { B } }  e.  U. ( R1 " On ) )
7 rankprb 8172 . . . . . 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 2507 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  { B }  e.  U. ( R1 " On ) )  ->  ( rank ` 
<< A ,  B >> )  =  suc  ( (
rank `  { A } )  u.  ( rank `  { A ,  { B } } ) ) )
10 snsspr1 4133 . . . . . . . 8  |-  { A }  C_  { A ,  { B } }
11 ssequn1 3637 . . . . . . . 8  |-  ( { A }  C_  { A ,  { B } }  <->  ( { A }  u.  { A ,  { B } } )  =  { A ,  { B } } )
1210, 11mpbi 208 . . . . . . 7  |-  ( { A }  u.  { A ,  { B } } )  =  { A ,  { B } }
1312fveq2i 5805 . . . . . 6  |-  ( rank `  ( { A }  u.  { A ,  { B } } ) )  =  ( rank `  { A ,  { B } } )
14 rankunb 8171 . . . . . . 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 8172 . . . . . 6  |-  ( ( A  e.  U. ( R1 " On )  /\  { B }  e.  U. ( R1 " On ) )  ->  ( rank `  { A ,  { B } } )  =  suc  ( ( rank `  A )  u.  ( rank `  { B }
) ) )
1713, 15, 163eqtr3a 2519 . . . . 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 4895 . . . . 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 2495 . . 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 8148 . . . . 5  |-  ( B  e.  U. ( R1
" On )  -> 
( rank `  { B } )  =  suc  ( rank `  B )
)
2322uneq2d 3621 . . . 4  |-  ( B  e.  U. ( R1
" On )  -> 
( ( rank `  A
)  u.  ( rank `  { B } ) )  =  ( (
rank `  A )  u.  suc  ( rank `  B
) ) )
24 suceq 4895 . . . 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 4895 . . . 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 2495 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 1370    e. wcel 1758    u. cun 3437    C_ wss 3439   {csn 3988   {cpr 3990   U.cuni 4202   Oncon0 4830   suc csuc 4832   "cima 4954   ` cfv 5529   R1cr1 8083   rankcrnk 8084   <<caltop 28151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-om 6590  df-recs 6945  df-rdg 6979  df-r1 8085  df-rank 8086  df-altop 28153
This theorem is referenced by: (None)
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