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Theorem hashbc0 13143
Description: The set of subsets of size zero is the singleton of the empty set. (Contributed by Mario Carneiro, 22-Apr-2015.)
Hypothesis
Ref Expression
ramval.c  |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } )
Assertion
Ref Expression
hashbc0  |-  ( A  e.  V  ->  ( A C 0 )  =  { (/) } )
Distinct variable groups:    a, b,
i    A, a, i
Allowed substitution hints:    A( b)    C( i, a, b)    V( i, a, b)

Proof of Theorem hashbc0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 0nn0 10069 . . 3  |-  0  e.  NN0
2 ramval.c . . . 4  |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } )
32hashbcval 13140 . . 3  |-  ( ( A  e.  V  /\  0  e.  NN0 )  -> 
( A C 0 )  =  { x  e.  ~P A  |  (
# `  x )  =  0 } )
41, 3mpan2 652 . 2  |-  ( A  e.  V  ->  ( A C 0 )  =  { x  e.  ~P A  |  ( # `  x
)  =  0 } )
5 vex 2867 . . . . . . 7  |-  x  e. 
_V
6 hasheq0 11443 . . . . . . 7  |-  ( x  e.  _V  ->  (
( # `  x )  =  0  <->  x  =  (/) ) )
75, 6ax-mp 8 . . . . . 6  |-  ( (
# `  x )  =  0  <->  x  =  (/) )
87anbi2i 675 . . . . 5  |-  ( ( x  e.  ~P A  /\  ( # `  x
)  =  0 )  <-> 
( x  e.  ~P A  /\  x  =  (/) ) )
9 id 19 . . . . . . 7  |-  ( x  =  (/)  ->  x  =  (/) )
10 0elpw 4259 . . . . . . 7  |-  (/)  e.  ~P A
119, 10syl6eqel 2446 . . . . . 6  |-  ( x  =  (/)  ->  x  e. 
~P A )
1211pm4.71ri 614 . . . . 5  |-  ( x  =  (/)  <->  ( x  e. 
~P A  /\  x  =  (/) ) )
138, 12bitr4i 243 . . . 4  |-  ( ( x  e.  ~P A  /\  ( # `  x
)  =  0 )  <-> 
x  =  (/) )
1413abbii 2470 . . 3  |-  { x  |  ( x  e. 
~P A  /\  ( # `
 x )  =  0 ) }  =  { x  |  x  =  (/) }
15 df-rab 2628 . . 3  |-  { x  e.  ~P A  |  (
# `  x )  =  0 }  =  { x  |  (
x  e.  ~P A  /\  ( # `  x
)  =  0 ) }
16 df-sn 3722 . . 3  |-  { (/) }  =  { x  |  x  =  (/) }
1714, 15, 163eqtr4i 2388 . 2  |-  { x  e.  ~P A  |  (
# `  x )  =  0 }  =  { (/) }
184, 17syl6eq 2406 1  |-  ( A  e.  V  ->  ( A C 0 )  =  { (/) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710   {cab 2344   {crab 2623   _Vcvv 2864   (/)c0 3531   ~Pcpw 3701   {csn 3716   ` cfv 5334  (class class class)co 5942    e. cmpt2 5944   0cc0 8824   NN0cn0 10054   #chash 11427
This theorem is referenced by:  0ram  13158
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-card 7659  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-nn 9834  df-n0 10055  df-z 10114  df-uz 10320  df-fz 10872  df-hash 11428
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