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Theorem hashrabsn01 30203
Description: The size of a restricted class abstraction restricted to a singleton is either 0 or 1. (Contributed by Alexander van der Vekens, 3-Sep-2018.)
Assertion
Ref Expression
hashrabsn01  |-  ( (
# `  { x  e.  { A }  |  ph } )  =  N  ->  ( N  =  0  \/  N  =  1 ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    N( x)

Proof of Theorem hashrabsn01
StepHypRef Expression
1 eqid 2438 . 2  |-  { x  e.  { A }  |  ph }  =  { x  e.  { A }  |  ph }
2 rabrsn 3940 . 2  |-  ( { x  e.  { A }  |  ph }  =  { x  e.  { A }  |  ph }  ->  ( { x  e.  { A }  |  ph }  =  (/)  \/  { x  e.  { A }  |  ph }  =  { A } ) )
3 fveq2 5686 . . . . 5  |-  ( { x  e.  { A }  |  ph }  =  (/) 
->  ( # `  {
x  e.  { A }  |  ph } )  =  ( # `  (/) ) )
43eqeq1d 2446 . . . 4  |-  ( { x  e.  { A }  |  ph }  =  (/) 
->  ( ( # `  {
x  e.  { A }  |  ph } )  =  N  <->  ( # `  (/) )  =  N ) )
5 eqcom 2440 . . . . . . 7  |-  ( (
# `  (/) )  =  N  <->  N  =  ( # `
 (/) ) )
65biimpi 194 . . . . . 6  |-  ( (
# `  (/) )  =  N  ->  N  =  ( # `  (/) ) )
7 hash0 12127 . . . . . 6  |-  ( # `  (/) )  =  0
86, 7syl6eq 2486 . . . . 5  |-  ( (
# `  (/) )  =  N  ->  N  = 
0 )
98orcd 392 . . . 4  |-  ( (
# `  (/) )  =  N  ->  ( N  =  0  \/  N  =  1 ) )
104, 9syl6bi 228 . . 3  |-  ( { x  e.  { A }  |  ph }  =  (/) 
->  ( ( # `  {
x  e.  { A }  |  ph } )  =  N  ->  ( N  =  0  \/  N  =  1 ) ) )
11 fveq2 5686 . . . . 5  |-  ( { x  e.  { A }  |  ph }  =  { A }  ->  ( # `
 { x  e. 
{ A }  |  ph } )  =  (
# `  { A } ) )
1211eqeq1d 2446 . . . 4  |-  ( { x  e.  { A }  |  ph }  =  { A }  ->  (
( # `  { x  e.  { A }  |  ph } )  =  N  <-> 
( # `  { A } )  =  N ) )
13 eqcom 2440 . . . . . . . . 9  |-  ( (
# `  { A } )  =  N  <-> 
N  =  ( # `  { A } ) )
1413biimpi 194 . . . . . . . 8  |-  ( (
# `  { A } )  =  N  ->  N  =  (
# `  { A } ) )
15 hashsng 12128 . . . . . . . 8  |-  ( A  e.  _V  ->  ( # `
 { A }
)  =  1 )
1614, 15sylan9eqr 2492 . . . . . . 7  |-  ( ( A  e.  _V  /\  ( # `  { A } )  =  N )  ->  N  = 
1 )
1716olcd 393 . . . . . 6  |-  ( ( A  e.  _V  /\  ( # `  { A } )  =  N )  ->  ( N  =  0  \/  N  =  1 ) )
1817ex 434 . . . . 5  |-  ( A  e.  _V  ->  (
( # `  { A } )  =  N  ->  ( N  =  0  \/  N  =  1 ) ) )
19 snprc 3934 . . . . . 6  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
20 fveq2 5686 . . . . . . . 8  |-  ( { A }  =  (/)  ->  ( # `  { A } )  =  (
# `  (/) ) )
2120eqeq1d 2446 . . . . . . 7  |-  ( { A }  =  (/)  ->  ( ( # `  { A } )  =  N  <-> 
( # `  (/) )  =  N ) )
2221, 9syl6bi 228 . . . . . 6  |-  ( { A }  =  (/)  ->  ( ( # `  { A } )  =  N  ->  ( N  =  0  \/  N  =  1 ) ) )
2319, 22sylbi 195 . . . . 5  |-  ( -.  A  e.  _V  ->  ( ( # `  { A } )  =  N  ->  ( N  =  0  \/  N  =  1 ) ) )
2418, 23pm2.61i 164 . . . 4  |-  ( (
# `  { A } )  =  N  ->  ( N  =  0  \/  N  =  1 ) )
2512, 24syl6bi 228 . . 3  |-  ( { x  e.  { A }  |  ph }  =  { A }  ->  (
( # `  { x  e.  { A }  |  ph } )  =  N  ->  ( N  =  0  \/  N  =  1 ) ) )
2610, 25jaoi 379 . 2  |-  ( ( { x  e.  { A }  |  ph }  =  (/)  \/  { x  e.  { A }  |  ph }  =  { A } )  ->  (
( # `  { x  e.  { A }  |  ph } )  =  N  ->  ( N  =  0  \/  N  =  1 ) ) )
271, 2, 26mp2b 10 1  |-  ( (
# `  { x  e.  { A }  |  ph } )  =  N  ->  ( N  =  0  \/  N  =  1 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   {crab 2714   _Vcvv 2967   (/)c0 3632   {csn 3872   ` cfv 5413   0cc0 9274   1c1 9275   #chash 12095
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 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-hash 12096
This theorem is referenced by:  rusgrasn  30528
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