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Theorem rankopb 7738
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 3946 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  <. A ,  B >.  =  { { A } ,  { A ,  B } } )
21fveq2d 5695 . 2  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  <. A ,  B >. )  =  ( rank `  { { A } ,  { A ,  B } } ) )
3 snwf 7695 . . . 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 7697 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  { A ,  B }  e.  U. ( R1 " On ) )
6 rankprb 7737 . . 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 3911 . . . . . 6  |-  { A }  C_  { A ,  B }
9 ssequn1 3481 . . . . . 6  |-  ( { A }  C_  { A ,  B }  <->  ( { A }  u.  { A ,  B } )  =  { A ,  B } )
108, 9mpbi 200 . . . . 5  |-  ( { A }  u.  { A ,  B }
)  =  { A ,  B }
1110fveq2i 5694 . . . 4  |-  ( rank `  ( { A }  u.  { A ,  B } ) )  =  ( rank `  { A ,  B }
)
12 rankunb 7736 . . . . 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 7737 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  { A ,  B }
)  =  suc  (
( rank `  A )  u.  ( rank `  B
) ) )
1511, 13, 143eqtr3a 2464 . . 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 4610 . . 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 2444 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 3282    C_ wss 3284   {csn 3778   {cpr 3779   <.cop 3781   U.cuni 3979   Oncon0 4545   suc csuc 4547   "cima 4844   ` cfv 5417   R1cr1 7648   rankcrnk 7649
This theorem is referenced by:  rankop  7744
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 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-recs 6596  df-rdg 6631  df-r1 7650  df-rank 7651
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