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Theorem rankopb 7519
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 3795 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  <. A ,  B >.  =  { { A } ,  { A ,  B } } )
21fveq2d 5489 . 2  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  <. A ,  B >. )  =  ( rank `  { { A } ,  { A ,  B } } ) )
3 snwf 7476 . . . 4  |-  ( A  e.  U. ( R1
" On )  ->  { A }  e.  U. ( R1 " On ) )
43adantr 453 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  { A }  e.  U. ( R1 " On ) )
5 prwf 7478 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  { A ,  B }  e.  U. ( R1 " On ) )
6 rankprb 7518 . . 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 644 . 2  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  { { A } ,  { A ,  B } } )  =  suc  ( ( rank `  { A } )  u.  ( rank `  { A ,  B } ) ) )
8 snsspr1 3765 . . . . . 6  |-  { A }  C_  { A ,  B }
9 ssequn1 3346 . . . . . 6  |-  ( { A }  C_  { A ,  B }  <->  ( { A }  u.  { A ,  B } )  =  { A ,  B } )
108, 9mpbi 201 . . . . 5  |-  ( { A }  u.  { A ,  B }
)  =  { A ,  B }
1110fveq2i 5488 . . . 4  |-  ( rank `  ( { A }  u.  { A ,  B } ) )  =  ( rank `  { A ,  B }
)
12 rankunb 7517 . . . . 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 644 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  ( { A }  u.  { A ,  B }
) )  =  ( ( rank `  { A } )  u.  ( rank `  { A ,  B } ) ) )
14 rankprb 7518 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  { A ,  B }
)  =  suc  (
( rank `  A )  u.  ( rank `  B
) ) )
1511, 13, 143eqtr3a 2340 . . 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 4456 . . 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 2320 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 set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1624    e. wcel 1685    u. cun 3151    C_ wss 3153   {csn 3641   {cpr 3642   <.cop 3644   U.cuni 3828   Oncon0 4391   suc csuc 4393   "cima 4691   ` cfv 5221   R1cr1 7429   rankcrnk 7430
This theorem is referenced by:  rankop  7525
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-recs 6383  df-rdg 6418  df-r1 7431  df-rank 7432
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