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Theorem rankopb 8322
Description: The rank of an ordered pair. Part of Exercise 4 of [Kunen] p. 107. (Contributed by Mario Carneiro, 10-Jun-2013.)
Assertion
Ref Expression
rankopb  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  <. A ,  B >. )  =  suc  suc  ( ( rank `  A )  u.  ( rank `  B
) ) )

Proof of Theorem rankopb
StepHypRef Expression
1 dfopg 4188 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  <. A ,  B >.  =  { { A } ,  { A ,  B } } )
21fveq2d 5885 . 2  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  <. A ,  B >. )  =  ( rank `  { { A } ,  { A ,  B } } ) )
3 snwf 8279 . . . 4  |-  ( A  e.  U. ( R1
" On )  ->  { A }  e.  U. ( R1 " On ) )
43adantr 466 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  { A }  e.  U. ( R1 " On ) )
5 prwf 8281 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  { A ,  B }  e.  U. ( R1 " On ) )
6 rankprb 8321 . . 3  |-  ( ( { A }  e.  U. ( R1 " On )  /\  { A ,  B }  e.  U. ( R1 " On ) )  ->  ( rank `  { { A } ,  { A ,  B } } )  =  suc  ( ( rank `  { A } )  u.  ( rank `  { A ,  B } ) ) )
74, 5, 6syl2anc 665 . 2  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  { { A } ,  { A ,  B } } )  =  suc  ( ( rank `  { A } )  u.  ( rank `  { A ,  B } ) ) )
8 snsspr1 4152 . . . . . 6  |-  { A }  C_  { A ,  B }
9 ssequn1 3642 . . . . . 6  |-  ( { A }  C_  { A ,  B }  <->  ( { A }  u.  { A ,  B } )  =  { A ,  B } )
108, 9mpbi 211 . . . . 5  |-  ( { A }  u.  { A ,  B }
)  =  { A ,  B }
1110fveq2i 5884 . . . 4  |-  ( rank `  ( { A }  u.  { A ,  B } ) )  =  ( rank `  { A ,  B }
)
12 rankunb 8320 . . . . 5  |-  ( ( { A }  e.  U. ( R1 " On )  /\  { A ,  B }  e.  U. ( R1 " On ) )  ->  ( rank `  ( { A }  u.  { A ,  B }
) )  =  ( ( rank `  { A } )  u.  ( rank `  { A ,  B } ) ) )
134, 5, 12syl2anc 665 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  ( { A }  u.  { A ,  B }
) )  =  ( ( rank `  { A } )  u.  ( rank `  { A ,  B } ) ) )
14 rankprb 8321 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  { A ,  B }
)  =  suc  (
( rank `  A )  u.  ( rank `  B
) ) )
1511, 13, 143eqtr3a 2494 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( ( rank `  { A } )  u.  ( rank `  { A ,  B }
) )  =  suc  ( ( rank `  A
)  u.  ( rank `  B ) ) )
16 suceq 5507 . . 3  |-  ( ( ( 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
) ) )
1715, 16syl 17 . 2  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  suc  ( ( rank `  { A }
)  u.  ( rank `  { A ,  B } ) )  =  suc  suc  ( ( rank `  A )  u.  ( rank `  B
) ) )
182, 7, 173eqtrd 2474 1  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  <. A ,  B >. )  =  suc  suc  ( ( rank `  A )  u.  ( rank `  B
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870    u. cun 3440    C_ wss 3442   {csn 4002   {cpr 4004   <.cop 4008   U.cuni 4222   "cima 4857   Oncon0 5442   suc csuc 5444   ` cfv 5601   R1cr1 8232   rankcrnk 8233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-om 6707  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-r1 8234  df-rank 8235
This theorem is referenced by:  rankop  8328
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