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Theorem pr2pwpr 12183
Description: The set of subsets of a pair having length 2 is the set of the pair as singleton. (Contributed by AV, 9-Dec-2018.)
Assertion
Ref Expression
pr2pwpr  |-  ( ( A  e.  V  /\  B  e.  W  /\  A  =/=  B )  ->  { p  e.  ~P { A ,  B }  |  p  ~~  2o }  =  { { A ,  B } } )
Distinct variable groups:    A, p    B, p
Allowed substitution hints:    V( p)    W( p)

Proof of Theorem pr2pwpr
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 elpwi 3869 . . . . . . 7  |-  ( s  e.  ~P { A ,  B }  ->  s  C_ 
{ A ,  B } )
2 prfi 7586 . . . . . . . . 9  |-  { A ,  B }  e.  Fin
3 ssfi 7533 . . . . . . . . 9  |-  ( ( { A ,  B }  e.  Fin  /\  s  C_ 
{ A ,  B } )  ->  s  e.  Fin )
42, 3mpan 670 . . . . . . . 8  |-  ( s 
C_  { A ,  B }  ->  s  e. 
Fin )
5 hash2 12163 . . . . . . . . . . . . . 14  |-  ( # `  2o )  =  2
65eqcomi 2447 . . . . . . . . . . . . 13  |-  2  =  ( # `  2o )
76a1i 11 . . . . . . . . . . . 12  |-  ( s  e.  Fin  ->  2  =  ( # `  2o ) )
87eqeq2d 2454 . . . . . . . . . . 11  |-  ( s  e.  Fin  ->  (
( # `  s )  =  2  <->  ( # `  s
)  =  ( # `  2o ) ) )
9 2onn 7079 . . . . . . . . . . . . 13  |-  2o  e.  om
10 nnfi 7503 . . . . . . . . . . . . 13  |-  ( 2o  e.  om  ->  2o  e.  Fin )
119, 10ax-mp 5 . . . . . . . . . . . 12  |-  2o  e.  Fin
12 hashen 12118 . . . . . . . . . . . 12  |-  ( ( s  e.  Fin  /\  2o  e.  Fin )  -> 
( ( # `  s
)  =  ( # `  2o )  <->  s  ~~  2o ) )
1311, 12mpan2 671 . . . . . . . . . . 11  |-  ( s  e.  Fin  ->  (
( # `  s )  =  ( # `  2o ) 
<->  s  ~~  2o ) )
148, 13bitrd 253 . . . . . . . . . 10  |-  ( s  e.  Fin  ->  (
( # `  s )  =  2  <->  s  ~~  2o ) )
15 hash2pwpr 12182 . . . . . . . . . . . 12  |-  ( ( ( # `  s
)  =  2  /\  s  e.  ~P { A ,  B }
)  ->  s  =  { A ,  B }
)
1615a1d 25 . . . . . . . . . . 11  |-  ( ( ( # `  s
)  =  2  /\  s  e.  ~P { A ,  B }
)  ->  ( ( A  e.  V  /\  B  e.  W  /\  A  =/=  B )  -> 
s  =  { A ,  B } ) )
1716ex 434 . . . . . . . . . 10  |-  ( (
# `  s )  =  2  ->  (
s  e.  ~P { A ,  B }  ->  ( ( A  e.  V  /\  B  e.  W  /\  A  =/= 
B )  ->  s  =  { A ,  B } ) ) )
1814, 17syl6bir 229 . . . . . . . . 9  |-  ( s  e.  Fin  ->  (
s  ~~  2o  ->  ( s  e.  ~P { A ,  B }  ->  ( ( A  e.  V  /\  B  e.  W  /\  A  =/= 
B )  ->  s  =  { A ,  B } ) ) ) )
1918com23 78 . . . . . . . 8  |-  ( s  e.  Fin  ->  (
s  e.  ~P { A ,  B }  ->  ( s  ~~  2o  ->  ( ( A  e.  V  /\  B  e.  W  /\  A  =/= 
B )  ->  s  =  { A ,  B } ) ) ) )
204, 19syl 16 . . . . . . 7  |-  ( s 
C_  { A ,  B }  ->  ( s  e.  ~P { A ,  B }  ->  (
s  ~~  2o  ->  ( ( A  e.  V  /\  B  e.  W  /\  A  =/=  B
)  ->  s  =  { A ,  B }
) ) ) )
211, 20mpcom 36 . . . . . 6  |-  ( s  e.  ~P { A ,  B }  ->  (
s  ~~  2o  ->  ( ( A  e.  V  /\  B  e.  W  /\  A  =/=  B
)  ->  s  =  { A ,  B }
) ) )
2221imp 429 . . . . 5  |-  ( ( s  e.  ~P { A ,  B }  /\  s  ~~  2o )  ->  ( ( A  e.  V  /\  B  e.  W  /\  A  =/= 
B )  ->  s  =  { A ,  B } ) )
2322com12 31 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  A  =/=  B )  -> 
( ( s  e. 
~P { A ,  B }  /\  s  ~~  2o )  ->  s  =  { A ,  B } ) )
24 prex 4534 . . . . . . . . . . . . 13  |-  { A ,  B }  e.  _V
2524prid2 3984 . . . . . . . . . . . 12  |-  { A ,  B }  e.  { { B } ,  { A ,  B } }
2625a1i 11 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  B  e.  W  /\  A  =/=  B )  ->  { A ,  B }  e.  { { B } ,  { A ,  B } } )
2726olcd 393 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  B  e.  W  /\  A  =/=  B )  -> 
( { A ,  B }  e.  { (/) ,  { A } }  \/  { A ,  B }  e.  { { B } ,  { A ,  B } } ) )
28 elun 3497 . . . . . . . . . 10  |-  ( { A ,  B }  e.  ( { (/) ,  { A } }  u.  { { B } ,  { A ,  B } } )  <->  ( { A ,  B }  e.  { (/) ,  { A } }  \/  { A ,  B }  e.  { { B } ,  { A ,  B } } ) )
2927, 28sylibr 212 . . . . . . . . 9  |-  ( ( A  e.  V  /\  B  e.  W  /\  A  =/=  B )  ->  { A ,  B }  e.  ( { (/) ,  { A } }  u.  { { B } ,  { A ,  B } } ) )
30 pwpr 4087 . . . . . . . . 9  |-  ~P { A ,  B }  =  ( { (/) ,  { A } }  u.  { { B } ,  { A ,  B } } )
3129, 30syl6eleqr 2534 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W  /\  A  =/=  B )  ->  { A ,  B }  e.  ~P { A ,  B } )
3231adantr 465 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  A  =/=  B
)  /\  s  =  { A ,  B }
)  ->  { A ,  B }  e.  ~P { A ,  B }
)
33 eleq1 2503 . . . . . . . 8  |-  ( s  =  { A ,  B }  ->  ( s  e.  ~P { A ,  B }  <->  { A ,  B }  e.  ~P { A ,  B }
) )
3433adantl 466 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  A  =/=  B
)  /\  s  =  { A ,  B }
)  ->  ( s  e.  ~P { A ,  B }  <->  { A ,  B }  e.  ~P { A ,  B } ) )
3532, 34mpbird 232 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  A  =/=  B
)  /\  s  =  { A ,  B }
)  ->  s  e.  ~P { A ,  B } )
36 pr2nelem 8171 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W  /\  A  =/=  B )  ->  { A ,  B }  ~~  2o )
3736adantr 465 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  A  =/=  B
)  /\  s  =  { A ,  B }
)  ->  { A ,  B }  ~~  2o )
38 breq1 4295 . . . . . . . 8  |-  ( s  =  { A ,  B }  ->  ( s 
~~  2o  <->  { A ,  B }  ~~  2o ) )
3938adantl 466 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  A  =/=  B
)  /\  s  =  { A ,  B }
)  ->  ( s  ~~  2o  <->  { A ,  B }  ~~  2o ) )
4037, 39mpbird 232 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  A  =/=  B
)  /\  s  =  { A ,  B }
)  ->  s  ~~  2o )
4135, 40jca 532 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  A  =/=  B
)  /\  s  =  { A ,  B }
)  ->  ( s  e.  ~P { A ,  B }  /\  s  ~~  2o ) )
4241ex 434 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  A  =/=  B )  -> 
( s  =  { A ,  B }  ->  ( s  e.  ~P { A ,  B }  /\  s  ~~  2o ) ) )
4323, 42impbid 191 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  A  =/=  B )  -> 
( ( s  e. 
~P { A ,  B }  /\  s  ~~  2o )  <->  s  =  { A ,  B }
) )
44 breq1 4295 . . . 4  |-  ( p  =  s  ->  (
p  ~~  2o  <->  s  ~~  2o ) )
4544elrab 3117 . . 3  |-  ( s  e.  { p  e. 
~P { A ,  B }  |  p  ~~  2o }  <->  ( s  e.  ~P { A ,  B }  /\  s  ~~  2o ) )
46 elsn 3891 . . 3  |-  ( s  e.  { { A ,  B } }  <->  s  =  { A ,  B }
)
4743, 45, 463bitr4g 288 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  A  =/=  B )  -> 
( s  e.  {
p  e.  ~P { A ,  B }  |  p  ~~  2o }  <->  s  e.  { { A ,  B } } ) )
4847eqrdv 2441 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  A  =/=  B )  ->  { p  e.  ~P { A ,  B }  |  p  ~~  2o }  =  { { A ,  B } } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   {crab 2719    u. cun 3326    C_ wss 3328   (/)c0 3637   ~Pcpw 3860   {csn 3877   {cpr 3879   class class class wbr 4292   ` cfv 5418   omcom 6476   2oc2o 6914    ~~ cen 7307   Fincfn 7310   2c2 10371   #chash 12103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-hash 12104
This theorem is referenced by:  pmtrprfval  15993
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