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Theorem hash2pwpr 12479
Description: If the size of a subset of an unordered pair is 2, the subset is the pair itself. (Contributed by Alexander van der Vekens, 9-Dec-2018.)
Assertion
Ref Expression
hash2pwpr  |-  ( ( ( # `  P
)  =  2  /\  P  e.  ~P { X ,  Y }
)  ->  P  =  { X ,  Y }
)

Proof of Theorem hash2pwpr
StepHypRef Expression
1 pwpr 4241 . . . . 5  |-  ~P { X ,  Y }  =  ( { (/) ,  { X } }  u.  { { Y } ,  { X ,  Y } } )
21eleq2i 2545 . . . 4  |-  ( P  e.  ~P { X ,  Y }  <->  P  e.  ( { (/) ,  { X } }  u.  { { Y } ,  { X ,  Y } } ) )
3 elun 3645 . . . 4  |-  ( P  e.  ( { (/) ,  { X } }  u.  { { Y } ,  { X ,  Y } } )  <->  ( P  e.  { (/) ,  { X } }  \/  P  e.  { { Y } ,  { X ,  Y } } ) )
42, 3bitri 249 . . 3  |-  ( P  e.  ~P { X ,  Y }  <->  ( P  e.  { (/) ,  { X } }  \/  P  e.  { { Y } ,  { X ,  Y } } ) )
5 elpri 4047 . . . . 5  |-  ( P  e.  { (/) ,  { X } }  ->  ( P  =  (/)  \/  P  =  { X } ) )
6 elpri 4047 . . . . 5  |-  ( P  e.  { { Y } ,  { X ,  Y } }  ->  ( P  =  { Y }  \/  P  =  { X ,  Y }
) )
75, 6orim12i 516 . . . 4  |-  ( ( P  e.  { (/) ,  { X } }  \/  P  e.  { { Y } ,  { X ,  Y } } )  ->  ( ( P  =  (/)  \/  P  =  { X } )  \/  ( P  =  { Y }  \/  P  =  { X ,  Y } ) ) )
8 fveq2 5864 . . . . . . . 8  |-  ( P  =  (/)  ->  ( # `  P )  =  (
# `  (/) ) )
9 hash0 12399 . . . . . . . . . 10  |-  ( # `  (/) )  =  0
109eqeq2i 2485 . . . . . . . . 9  |-  ( (
# `  P )  =  ( # `  (/) )  <->  ( # `  P
)  =  0 )
11 eqeq1 2471 . . . . . . . . . 10  |-  ( (
# `  P )  =  0  ->  (
( # `  P )  =  2  <->  0  = 
2 ) )
12 2ne0 10624 . . . . . . . . . . . 12  |-  2  =/=  0
1312necomi 2737 . . . . . . . . . . 11  |-  0  =/=  2
14 eqneqall 2674 . . . . . . . . . . 11  |-  ( 0  =  2  ->  (
0  =/=  2  ->  P  =  { X ,  Y } ) )
1513, 14mpi 17 . . . . . . . . . 10  |-  ( 0  =  2  ->  P  =  { X ,  Y } )
1611, 15syl6bi 228 . . . . . . . . 9  |-  ( (
# `  P )  =  0  ->  (
( # `  P )  =  2  ->  P  =  { X ,  Y } ) )
1710, 16sylbi 195 . . . . . . . 8  |-  ( (
# `  P )  =  ( # `  (/) )  -> 
( ( # `  P
)  =  2  ->  P  =  { X ,  Y } ) )
188, 17syl 16 . . . . . . 7  |-  ( P  =  (/)  ->  ( (
# `  P )  =  2  ->  P  =  { X ,  Y } ) )
19 hashsng 12400 . . . . . . . . 9  |-  ( X  e.  _V  ->  ( # `
 { X }
)  =  1 )
20 fveq2 5864 . . . . . . . . . . . 12  |-  ( { X }  =  P  ->  ( # `  { X } )  =  (
# `  P )
)
2120eqcoms 2479 . . . . . . . . . . 11  |-  ( P  =  { X }  ->  ( # `  { X } )  =  (
# `  P )
)
2221eqeq1d 2469 . . . . . . . . . 10  |-  ( P  =  { X }  ->  ( ( # `  { X } )  =  1  <-> 
( # `  P )  =  1 ) )
23 eqeq1 2471 . . . . . . . . . . 11  |-  ( (
# `  P )  =  1  ->  (
( # `  P )  =  2  <->  1  = 
2 ) )
24 1ne2 10744 . . . . . . . . . . . 12  |-  1  =/=  2
25 eqneqall 2674 . . . . . . . . . . . 12  |-  ( 1  =  2  ->  (
1  =/=  2  ->  P  =  { X ,  Y } ) )
2624, 25mpi 17 . . . . . . . . . . 11  |-  ( 1  =  2  ->  P  =  { X ,  Y } )
2723, 26syl6bi 228 . . . . . . . . . 10  |-  ( (
# `  P )  =  1  ->  (
( # `  P )  =  2  ->  P  =  { X ,  Y } ) )
2822, 27syl6bi 228 . . . . . . . . 9  |-  ( P  =  { X }  ->  ( ( # `  { X } )  =  1  ->  ( ( # `  P )  =  2  ->  P  =  { X ,  Y }
) ) )
2919, 28syl5com 30 . . . . . . . 8  |-  ( X  e.  _V  ->  ( P  =  { X }  ->  ( ( # `  P )  =  2  ->  P  =  { X ,  Y }
) ) )
30 snprc 4091 . . . . . . . . 9  |-  ( -.  X  e.  _V  <->  { X }  =  (/) )
31 eqeq2 2482 . . . . . . . . . 10  |-  ( { X }  =  (/)  ->  ( P  =  { X }  <->  P  =  (/) ) )
328, 9syl6eq 2524 . . . . . . . . . . . 12  |-  ( P  =  (/)  ->  ( # `  P )  =  0 )
3332eqeq1d 2469 . . . . . . . . . . 11  |-  ( P  =  (/)  ->  ( (
# `  P )  =  2  <->  0  = 
2 ) )
3433, 15syl6bi 228 . . . . . . . . . 10  |-  ( P  =  (/)  ->  ( (
# `  P )  =  2  ->  P  =  { X ,  Y } ) )
3531, 34syl6bi 228 . . . . . . . . 9  |-  ( { X }  =  (/)  ->  ( P  =  { X }  ->  ( (
# `  P )  =  2  ->  P  =  { X ,  Y } ) ) )
3630, 35sylbi 195 . . . . . . . 8  |-  ( -.  X  e.  _V  ->  ( P  =  { X }  ->  ( ( # `  P )  =  2  ->  P  =  { X ,  Y }
) ) )
3729, 36pm2.61i 164 . . . . . . 7  |-  ( P  =  { X }  ->  ( ( # `  P
)  =  2  ->  P  =  { X ,  Y } ) )
3818, 37jaoi 379 . . . . . 6  |-  ( ( P  =  (/)  \/  P  =  { X } )  ->  ( ( # `  P )  =  2  ->  P  =  { X ,  Y }
) )
39 hashsng 12400 . . . . . . . . 9  |-  ( Y  e.  _V  ->  ( # `
 { Y }
)  =  1 )
40 fveq2 5864 . . . . . . . . . . . 12  |-  ( { Y }  =  P  ->  ( # `  { Y } )  =  (
# `  P )
)
4140eqcoms 2479 . . . . . . . . . . 11  |-  ( P  =  { Y }  ->  ( # `  { Y } )  =  (
# `  P )
)
4241eqeq1d 2469 . . . . . . . . . 10  |-  ( P  =  { Y }  ->  ( ( # `  { Y } )  =  1  <-> 
( # `  P )  =  1 ) )
4342, 27syl6bi 228 . . . . . . . . 9  |-  ( P  =  { Y }  ->  ( ( # `  { Y } )  =  1  ->  ( ( # `  P )  =  2  ->  P  =  { X ,  Y }
) ) )
4439, 43syl5com 30 . . . . . . . 8  |-  ( Y  e.  _V  ->  ( P  =  { Y }  ->  ( ( # `  P )  =  2  ->  P  =  { X ,  Y }
) ) )
45 snprc 4091 . . . . . . . . 9  |-  ( -.  Y  e.  _V  <->  { Y }  =  (/) )
46 eqeq2 2482 . . . . . . . . . 10  |-  ( { Y }  =  (/)  ->  ( P  =  { Y }  <->  P  =  (/) ) )
478eqeq1d 2469 . . . . . . . . . . 11  |-  ( P  =  (/)  ->  ( (
# `  P )  =  2  <->  ( # `  (/) )  =  2 ) )
489eqeq1i 2474 . . . . . . . . . . . 12  |-  ( (
# `  (/) )  =  2  <->  0  =  2 )
4948, 15sylbi 195 . . . . . . . . . . 11  |-  ( (
# `  (/) )  =  2  ->  P  =  { X ,  Y }
)
5047, 49syl6bi 228 . . . . . . . . . 10  |-  ( P  =  (/)  ->  ( (
# `  P )  =  2  ->  P  =  { X ,  Y } ) )
5146, 50syl6bi 228 . . . . . . . . 9  |-  ( { Y }  =  (/)  ->  ( P  =  { Y }  ->  ( (
# `  P )  =  2  ->  P  =  { X ,  Y } ) ) )
5245, 51sylbi 195 . . . . . . . 8  |-  ( -.  Y  e.  _V  ->  ( P  =  { Y }  ->  ( ( # `  P )  =  2  ->  P  =  { X ,  Y }
) ) )
5344, 52pm2.61i 164 . . . . . . 7  |-  ( P  =  { Y }  ->  ( ( # `  P
)  =  2  ->  P  =  { X ,  Y } ) )
54 ax-1 6 . . . . . . 7  |-  ( P  =  { X ,  Y }  ->  ( (
# `  P )  =  2  ->  P  =  { X ,  Y } ) )
5553, 54jaoi 379 . . . . . 6  |-  ( ( P  =  { Y }  \/  P  =  { X ,  Y }
)  ->  ( ( # `
 P )  =  2  ->  P  =  { X ,  Y }
) )
5638, 55jaoi 379 . . . . 5  |-  ( ( ( P  =  (/)  \/  P  =  { X } )  \/  ( P  =  { Y }  \/  P  =  { X ,  Y }
) )  ->  (
( # `  P )  =  2  ->  P  =  { X ,  Y } ) )
5756com12 31 . . . 4  |-  ( (
# `  P )  =  2  ->  (
( ( P  =  (/)  \/  P  =  { X } )  \/  ( P  =  { Y }  \/  P  =  { X ,  Y }
) )  ->  P  =  { X ,  Y } ) )
587, 57syl5 32 . . 3  |-  ( (
# `  P )  =  2  ->  (
( P  e.  { (/)
,  { X } }  \/  P  e.  { { Y } ,  { X ,  Y } } )  ->  P  =  { X ,  Y } ) )
594, 58syl5bi 217 . 2  |-  ( (
# `  P )  =  2  ->  ( P  e.  ~P { X ,  Y }  ->  P  =  { X ,  Y } ) )
6059imp 429 1  |-  ( ( ( # `  P
)  =  2  /\  P  e.  ~P { X ,  Y }
)  ->  P  =  { X ,  Y }
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113    u. cun 3474   (/)c0 3785   ~Pcpw 4010   {csn 4027   {cpr 4029   ` cfv 5586   0cc0 9488   1c1 9489   2c2 10581   #chash 12367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-hash 12368
This theorem is referenced by:  pr2pwpr  12480
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