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Theorem rankopb 7408
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 3694 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  <. A ,  B >.  =  { { A } ,  { A ,  B } } )
21fveq2d 5381 . 2  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  <. A ,  B >. )  =  ( rank `  { { A } ,  { A ,  B } } ) )
3 snwf 7365 . . . 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 7367 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  { A ,  B }  e.  U. ( R1 " On ) )
6 rankprb 7407 . . 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 645 . 2  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  { { A } ,  { A ,  B } } )  =  suc  ( ( rank `  { A } )  u.  ( rank `  { A ,  B } ) ) )
8 snsspr1 3664 . . . . . 6  |-  { A }  C_  { A ,  B }
9 ssequn1 3255 . . . . . 6  |-  ( { A }  C_  { A ,  B }  <->  ( { A }  u.  { A ,  B } )  =  { A ,  B } )
108, 9mpbi 201 . . . . 5  |-  ( { A }  u.  { A ,  B }
)  =  { A ,  B }
1110fveq2i 5380 . . . 4  |-  ( rank `  ( { A }  u.  { A ,  B } ) )  =  ( rank `  { A ,  B }
)
12 rankunb 7406 . . . . 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 645 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  ( { A }  u.  { A ,  B }
) )  =  ( ( rank `  { A } )  u.  ( rank `  { A ,  B } ) ) )
14 rankprb 7407 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  { A ,  B }
)  =  suc  (
( rank `  A )  u.  ( rank `  B
) ) )
1511, 13, 143eqtr3a 2309 . . 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 4350 . . 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 2289 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 1619    e. wcel 1621    u. cun 3076    C_ wss 3078   {csn 3544   {cpr 3545   <.cop 3547   U.cuni 3727   Oncon0 4285   suc csuc 4287   "cima 4583   ` cfv 4592   R1cr1 7318   rankcrnk 7319
This theorem is referenced by:  rankop  7414
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-recs 6274  df-rdg 6309  df-r1 7320  df-rank 7321
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