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Theorem pwf1oexmid 14752
Description: An exercise related to  N copies of a singleton and the power set of a singleton (where the latter can also be thought of as representing truth values). Posed as an exercise by Martin Escardo online. (Contributed by Jim Kingdon, 3-Sep-2023.)
Hypothesis
Ref Expression
pwle2.t  |-  T  = 
U_ x  e.  N  ( { x }  X.  1o )
Assertion
Ref Expression
pwf1oexmid  |-  ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  ->  ( ran  G  =  ~P 1o  <->  ( N  =  2o  /\ EXMID ) ) )
Distinct variable group:    x, N
Allowed substitution hints:    T( x)    G( x)

Proof of Theorem pwf1oexmid
StepHypRef Expression
1 pwle2.t . . . . . 6  |-  T  = 
U_ x  e.  N  ( { x }  X.  1o )
21pwle2 14751 . . . . 5  |-  ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  ->  N  C_  2o )
32adantr 276 . . . 4  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  ->  N  C_  2o )
4 pw1dom2 7226 . . . . . 6  |-  2o  ~<_  ~P 1o
5 iunxpconst 4687 . . . . . . . . . . . 12  |-  U_ x  e.  N  ( {
x }  X.  1o )  =  ( N  X.  1o )
6 df1o2 6430 . . . . . . . . . . . . 13  |-  1o  =  { (/) }
76xpeq2i 4648 . . . . . . . . . . . 12  |-  ( N  X.  1o )  =  ( N  X.  { (/)
} )
81, 5, 73eqtri 2202 . . . . . . . . . . 11  |-  T  =  ( N  X.  { (/)
} )
9 peano1 4594 . . . . . . . . . . . 12  |-  (/)  e.  om
10 xpsneng 6822 . . . . . . . . . . . 12  |-  ( ( N  e.  om  /\  (/) 
e.  om )  ->  ( N  X.  { (/) } ) 
~~  N )
119, 10mpan2 425 . . . . . . . . . . 11  |-  ( N  e.  om  ->  ( N  X.  { (/) } ) 
~~  N )
128, 11eqbrtrid 4039 . . . . . . . . . 10  |-  ( N  e.  om  ->  T  ~~  N )
1312ad2antrr 488 . . . . . . . . 9  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  ->  T  ~~  N )
1413ensymd 6783 . . . . . . . 8  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  ->  N  ~~  T )
15 relen 6744 . . . . . . . . . 10  |-  Rel  ~~
16 brrelex1 4666 . . . . . . . . . 10  |-  ( ( Rel  ~~  /\  T  ~~  N )  ->  T  e.  _V )
1715, 13, 16sylancr 414 . . . . . . . . 9  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  ->  T  e.  _V )
18 simplr 528 . . . . . . . . . 10  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  ->  G : T -1-1-> ~P 1o )
19 simpr 110 . . . . . . . . . 10  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  ->  ran  G  =  ~P 1o )
20 dff1o5 5471 . . . . . . . . . 10  |-  ( G : T -1-1-onto-> ~P 1o  <->  ( G : T -1-1-> ~P 1o  /\  ran  G  =  ~P 1o ) )
2118, 19, 20sylanbrc 417 . . . . . . . . 9  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  ->  G : T -1-1-onto-> ~P 1o )
22 f1oeng 6757 . . . . . . . . 9  |-  ( ( T  e.  _V  /\  G : T -1-1-onto-> ~P 1o )  ->  T  ~~  ~P 1o )
2317, 21, 22syl2anc 411 . . . . . . . 8  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  ->  T  ~~  ~P 1o )
24 entr 6784 . . . . . . . 8  |-  ( ( N  ~~  T  /\  T  ~~  ~P 1o )  ->  N  ~~  ~P 1o )
2514, 23, 24syl2anc 411 . . . . . . 7  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  ->  N  ~~  ~P 1o )
2625ensymd 6783 . . . . . 6  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  ->  ~P 1o  ~~  N )
27 domentr 6791 . . . . . 6  |-  ( ( 2o  ~<_  ~P 1o  /\  ~P 1o  ~~  N )  ->  2o 
~<_  N )
284, 26, 27sylancr 414 . . . . 5  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  ->  2o 
~<_  N )
29 2onn 6522 . . . . . . 7  |-  2o  e.  om
30 nndomo 6864 . . . . . . 7  |-  ( ( 2o  e.  om  /\  N  e.  om )  ->  ( 2o  ~<_  N  <->  2o  C_  N
) )
3129, 30mpan 424 . . . . . 6  |-  ( N  e.  om  ->  ( 2o 
~<_  N  <->  2o  C_  N ) )
3231ad2antrr 488 . . . . 5  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  -> 
( 2o  ~<_  N  <->  2o  C_  N
) )
3328, 32mpbid 147 . . . 4  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  ->  2o  C_  N )
343, 33eqssd 3173 . . 3  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  ->  N  =  2o )
3526, 34breqtrd 4030 . . . 4  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  ->  ~P 1o  ~~  2o )
36 exmidpw 6908 . . . 4  |-  (EXMID  <->  ~P 1o  ~~  2o )
3735, 36sylibr 134 . . 3  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  -> EXMID )
3834, 37jca 306 . 2  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  -> 
( N  =  2o 
/\ EXMID
) )
39 simplr 528 . . . . 5  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ( N  =  2o  /\ EXMID ) )  ->  G : T -1-1-> ~P 1o )
4012ad2antrr 488 . . . . . . . 8  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ( N  =  2o  /\ EXMID ) )  ->  T  ~~  N )
41 simprl 529 . . . . . . . 8  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ( N  =  2o  /\ EXMID ) )  ->  N  =  2o )
4240, 41breqtrd 4030 . . . . . . 7  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ( N  =  2o  /\ EXMID ) )  ->  T  ~~  2o )
43 simprr 531 . . . . . . . . 9  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ( N  =  2o  /\ EXMID ) )  -> EXMID )
4443, 36sylib 122 . . . . . . . 8  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ( N  =  2o  /\ EXMID ) )  ->  ~P 1o  ~~  2o )
4544ensymd 6783 . . . . . . 7  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ( N  =  2o  /\ EXMID ) )  ->  2o  ~~  ~P 1o )
46 entr 6784 . . . . . . 7  |-  ( ( T  ~~  2o  /\  2o  ~~  ~P 1o )  ->  T  ~~  ~P 1o )
4742, 45, 46syl2anc 411 . . . . . 6  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ( N  =  2o  /\ EXMID ) )  ->  T  ~~  ~P 1o )
48 nnfi 6872 . . . . . . . 8  |-  ( 2o  e.  om  ->  2o  e.  Fin )
4929, 48mp1i 10 . . . . . . 7  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ( N  =  2o  /\ EXMID ) )  ->  2o  e.  Fin )
50 enfi 6873 . . . . . . . 8  |-  ( ~P 1o  ~~  2o  ->  ( ~P 1o  e.  Fin  <->  2o  e.  Fin ) )
5144, 50syl 14 . . . . . . 7  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ( N  =  2o  /\ EXMID ) )  -> 
( ~P 1o  e.  Fin 
<->  2o  e.  Fin )
)
5249, 51mpbird 167 . . . . . 6  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ( N  =  2o  /\ EXMID ) )  ->  ~P 1o  e.  Fin )
53 f1finf1o 6946 . . . . . 6  |-  ( ( T  ~~  ~P 1o  /\ 
~P 1o  e.  Fin )  ->  ( G : T -1-1-> ~P 1o  <->  G : T
-1-1-onto-> ~P 1o ) )
5447, 52, 53syl2anc 411 . . . . 5  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ( N  =  2o  /\ EXMID ) )  -> 
( G : T -1-1-> ~P 1o  <->  G : T -1-1-onto-> ~P 1o ) )
5539, 54mpbid 147 . . . 4  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ( N  =  2o  /\ EXMID ) )  ->  G : T -1-1-onto-> ~P 1o )
5655, 20sylib 122 . . 3  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ( N  =  2o  /\ EXMID ) )  -> 
( G : T -1-1-> ~P 1o  /\  ran  G  =  ~P 1o ) )
5756simprd 114 . 2  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ( N  =  2o  /\ EXMID ) )  ->  ran  G  =  ~P 1o )
5838, 57impbida 596 1  |-  ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  ->  ( ran  G  =  ~P 1o  <->  ( N  =  2o  /\ EXMID ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   _Vcvv 2738    C_ wss 3130   (/)c0 3423   ~Pcpw 3576   {csn 3593   U_ciun 3887   class class class wbr 4004  EXMIDwem 4195   omcom 4590    X. cxp 4625   ran crn 4628   Rel wrel 4632   -1-1->wf1 5214   -1-1-onto->wf1o 5216   1oc1o 6410   2oc2o 6411    ~~ cen 6738    ~<_ cdom 6739   Fincfn 6740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-exmid 4196  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-1o 6417  df-2o 6418  df-er 6535  df-en 6741  df-dom 6742  df-fin 6743
This theorem is referenced by: (None)
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