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Theorem pr2pwpr 12524
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 4024 . . . . . . 7  |-  ( s  e.  ~P { A ,  B }  ->  s  C_ 
{ A ,  B } )
2 prfi 7813 . . . . . . . . 9  |-  { A ,  B }  e.  Fin
3 ssfi 7759 . . . . . . . . 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 12474 . . . . . . . . . . . . . 14  |-  ( # `  2o )  =  2
65eqcomi 2470 . . . . . . . . . . . . 13  |-  2  =  ( # `  2o )
76a1i 11 . . . . . . . . . . . 12  |-  ( s  e.  Fin  ->  2  =  ( # `  2o ) )
87eqeq2d 2471 . . . . . . . . . . 11  |-  ( s  e.  Fin  ->  (
( # `  s )  =  2  <->  ( # `  s
)  =  ( # `  2o ) ) )
9 2onn 7307 . . . . . . . . . . . . 13  |-  2o  e.  om
10 nnfi 7729 . . . . . . . . . . . . 13  |-  ( 2o  e.  om  ->  2o  e.  Fin )
119, 10ax-mp 5 . . . . . . . . . . . 12  |-  2o  e.  Fin
12 hashen 12423 . . . . . . . . . . . 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 12523 . . . . . . . . . . . 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 4698 . . . . . . . . . . . . 13  |-  { A ,  B }  e.  _V
2524prid2 4141 . . . . . . . . . . . 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 3641 . . . . . . . . . 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 4247 . . . . . . . . 9  |-  ~P { A ,  B }  =  ( { (/) ,  { A } }  u.  { { B } ,  { A ,  B } } )
3129, 30syl6eleqr 2556 . . . . . . . 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 2529 . . . . . . . 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 8399 . . . . . . . 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 4459 . . . . . . . 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 4459 . . . 4  |-  ( p  =  s  ->  (
p  ~~  2o  <->  s  ~~  2o ) )
4544elrab 3257 . . 3  |-  ( s  e.  { p  e. 
~P { A ,  B }  |  p  ~~  2o }  <->  ( s  e.  ~P { A ,  B }  /\  s  ~~  2o ) )
46 elsn 4046 . . 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 2454 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 973    = wceq 1395    e. wcel 1819    =/= wne 2652   {crab 2811    u. cun 3469    C_ wss 3471   (/)c0 3793   ~Pcpw 4015   {csn 4032   {cpr 4034   class class class wbr 4456   ` cfv 5594   omcom 6699   2oc2o 7142    ~~ cen 7532   Fincfn 7535   2c2 10606   #chash 12408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-hash 12409
This theorem is referenced by:  pmtrprfval  16639
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