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Theorem rankprb 7407
Description: The rank of an unordered pair. Part of Exercise 30 of [Enderton] p. 207. (Contributed by Mario Carneiro, 10-Jun-2013.)
Assertion
Ref Expression
rankprb  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  { A ,  B }
)  =  suc  (
( rank `  A )  u.  ( rank `  B
) ) )

Proof of Theorem rankprb
StepHypRef Expression
1 snwf 7365 . . . 4  |-  ( A  e.  U. ( R1
" On )  ->  { A }  e.  U. ( R1 " On ) )
2 snwf 7365 . . . 4  |-  ( B  e.  U. ( R1
" On )  ->  { B }  e.  U. ( R1 " On ) )
3 rankunb 7406 . . . 4  |-  ( ( { A }  e.  U. ( R1 " On )  /\  { B }  e.  U. ( R1 " On ) )  ->  ( rank `  ( { A }  u.  { B } ) )  =  ( ( rank `  { A } )  u.  ( rank `  { B }
) ) )
41, 2, 3syl2an 465 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  ( { A }  u.  { B } ) )  =  ( ( rank `  { A } )  u.  ( rank `  { B }
) ) )
5 ranksnb 7383 . . . 4  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  { A } )  =  suc  ( rank `  A )
)
6 ranksnb 7383 . . . 4  |-  ( B  e.  U. ( R1
" On )  -> 
( rank `  { B } )  =  suc  ( rank `  B )
)
7 uneq12 3234 . . . 4  |-  ( ( ( rank `  { A } )  =  suc  ( rank `  A )  /\  ( rank `  { B } )  =  suc  ( rank `  B )
)  ->  ( ( rank `  { A }
)  u.  ( rank `  { B } ) )  =  ( suc  ( rank `  A
)  u.  suc  ( rank `  B ) ) )
85, 6, 7syl2an 465 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( ( rank `  { A } )  u.  ( rank `  { B } ) )  =  ( suc  ( rank `  A )  u.  suc  ( rank `  B )
) )
94, 8eqtrd 2285 . 2  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  ( { A }  u.  { B } ) )  =  ( suc  ( rank `  A )  u.  suc  ( rank `  B )
) )
10 df-pr 3551 . . 3  |-  { A ,  B }  =  ( { A }  u.  { B } )
1110fveq2i 5380 . 2  |-  ( rank `  { A ,  B } )  =  (
rank `  ( { A }  u.  { B } ) )
12 rankon 7351 . . . 4  |-  ( rank `  A )  e.  On
1312onordi 4388 . . 3  |-  Ord  ( rank `  A )
14 rankon 7351 . . . 4  |-  ( rank `  B )  e.  On
1514onordi 4388 . . 3  |-  Ord  ( rank `  B )
16 ordsucun 4507 . . 3  |-  ( ( Ord  ( rank `  A
)  /\  Ord  ( rank `  B ) )  ->  suc  ( ( rank `  A
)  u.  ( rank `  B ) )  =  ( suc  ( rank `  A )  u.  suc  ( rank `  B )
) )
1713, 15, 16mp2an 656 . 2  |-  suc  (
( rank `  A )  u.  ( rank `  B
) )  =  ( suc  ( rank `  A
)  u.  suc  ( rank `  B ) )
189, 11, 173eqtr4g 2310 1  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) )  ->  ( rank `  { A ,  B }
)  =  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   {csn 3544   {cpr 3545   U.cuni 3727   Ord word 4284   Oncon0 4285   suc csuc 4287   "cima 4583   ` cfv 4592   R1cr1 7318   rankcrnk 7319
This theorem is referenced by:  rankopb  7408  rankpr  7413  r1limwun  8238  rankaltopb  23687
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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