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Theorem rankopb 7734
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 3942 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  <. A ,  B >.  =  { { A } ,  { A ,  B } } )
21fveq2d 5691 . 2  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  <. A ,  B >. )  =  ( rank `  { { A } ,  { A ,  B } } ) )
3 snwf 7691 . . . 4  |-  ( A  e.  U. ( R1
" On )  ->  { A }  e.  U. ( R1 " On ) )
43adantr 452 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  { A }  e.  U. ( R1 " On ) )
5 prwf 7693 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  { A ,  B }  e.  U. ( R1 " On ) )
6 rankprb 7733 . . 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 643 . 2  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  { { A } ,  { A ,  B } } )  =  suc  ( ( rank `  { A } )  u.  ( rank `  { A ,  B } ) ) )
8 snsspr1 3907 . . . . . 6  |-  { A }  C_  { A ,  B }
9 ssequn1 3477 . . . . . 6  |-  ( { A }  C_  { A ,  B }  <->  ( { A }  u.  { A ,  B } )  =  { A ,  B } )
108, 9mpbi 200 . . . . 5  |-  ( { A }  u.  { A ,  B }
)  =  { A ,  B }
1110fveq2i 5690 . . . 4  |-  ( rank `  ( { A }  u.  { A ,  B } ) )  =  ( rank `  { A ,  B }
)
12 rankunb 7732 . . . . 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 643 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  ( { A }  u.  { A ,  B }
) )  =  ( ( rank `  { A } )  u.  ( rank `  { A ,  B } ) ) )
14 rankprb 7733 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  { A ,  B }
)  =  suc  (
( rank `  A )  u.  ( rank `  B
) ) )
1511, 13, 143eqtr3a 2460 . . 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 4606 . . 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 16 . 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 2440 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 4    /\ wa 359    = wceq 1649    e. wcel 1721    u. cun 3278    C_ wss 3280   {csn 3774   {cpr 3775   <.cop 3777   U.cuni 3975   Oncon0 4541   suc csuc 4543   "cima 4840   ` cfv 5413   R1cr1 7644   rankcrnk 7645
This theorem is referenced by:  rankop  7740
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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