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Theorem hashv01gt1 12421
Description: The size of a set is either 0 or 1 or greater than 1. (Contributed by Alexander van der Vekens, 29-Dec-2017.)
Assertion
Ref Expression
hashv01gt1  |-  ( M  e.  V  ->  (
( # `  M )  =  0  \/  ( # `
 M )  =  1  \/  1  < 
( # `  M ) ) )

Proof of Theorem hashv01gt1
StepHypRef Expression
1 hashnn0pnf 12418 . 2  |-  ( M  e.  V  ->  (
( # `  M )  e.  NN0  \/  ( # `
 M )  = +oo ) )
2 elnn0 10818 . . . 4  |-  ( (
# `  M )  e.  NN0  <->  ( ( # `  M )  e.  NN  \/  ( # `  M
)  =  0 ) )
3 exmidne 2662 . . . . . . . 8  |-  ( (
# `  M )  =  1  \/  ( # `
 M )  =/=  1 )
4 nngt1ne1 10583 . . . . . . . . 9  |-  ( (
# `  M )  e.  NN  ->  ( 1  <  ( # `  M
)  <->  ( # `  M
)  =/=  1 ) )
54orbi2d 701 . . . . . . . 8  |-  ( (
# `  M )  e.  NN  ->  ( (
( # `  M )  =  1  \/  1  <  ( # `  M
) )  <->  ( ( # `
 M )  =  1  \/  ( # `  M )  =/=  1
) ) )
63, 5mpbiri 233 . . . . . . 7  |-  ( (
# `  M )  e.  NN  ->  ( ( # `
 M )  =  1  \/  1  < 
( # `  M ) ) )
76olcd 393 . . . . . 6  |-  ( (
# `  M )  e.  NN  ->  ( ( # `
 M )  =  0  \/  ( (
# `  M )  =  1  \/  1  <  ( # `  M
) ) ) )
8 3orass 976 . . . . . 6  |-  ( ( ( # `  M
)  =  0  \/  ( # `  M
)  =  1  \/  1  <  ( # `  M ) )  <->  ( ( # `
 M )  =  0  \/  ( (
# `  M )  =  1  \/  1  <  ( # `  M
) ) ) )
97, 8sylibr 212 . . . . 5  |-  ( (
# `  M )  e.  NN  ->  ( ( # `
 M )  =  0  \/  ( # `  M )  =  1  \/  1  <  ( # `
 M ) ) )
10 3mix1 1165 . . . . 5  |-  ( (
# `  M )  =  0  ->  (
( # `  M )  =  0  \/  ( # `
 M )  =  1  \/  1  < 
( # `  M ) ) )
119, 10jaoi 379 . . . 4  |-  ( ( ( # `  M
)  e.  NN  \/  ( # `  M )  =  0 )  -> 
( ( # `  M
)  =  0  \/  ( # `  M
)  =  1  \/  1  <  ( # `  M ) ) )
122, 11sylbi 195 . . 3  |-  ( (
# `  M )  e.  NN0  ->  ( ( # `
 M )  =  0  \/  ( # `  M )  =  1  \/  1  <  ( # `
 M ) ) )
13 1re 9612 . . . . . 6  |-  1  e.  RR
14 ltpnf 11356 . . . . . 6  |-  ( 1  e.  RR  ->  1  < +oo )
1513, 14ax-mp 5 . . . . 5  |-  1  < +oo
16 breq2 4460 . . . . 5  |-  ( (
# `  M )  = +oo  ->  ( 1  <  ( # `  M
)  <->  1  < +oo ) )
1715, 16mpbiri 233 . . . 4  |-  ( (
# `  M )  = +oo  ->  1  <  (
# `  M )
)
18173mix3d 1173 . . 3  |-  ( (
# `  M )  = +oo  ->  ( ( # `
 M )  =  0  \/  ( # `  M )  =  1  \/  1  <  ( # `
 M ) ) )
1912, 18jaoi 379 . 2  |-  ( ( ( # `  M
)  e.  NN0  \/  ( # `  M )  = +oo )  -> 
( ( # `  M
)  =  0  \/  ( # `  M
)  =  1  \/  1  <  ( # `  M ) ) )
201, 19syl 16 1  |-  ( M  e.  V  ->  (
( # `  M )  =  0  \/  ( # `
 M )  =  1  \/  1  < 
( # `  M ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    \/ w3o 972    = wceq 1395    e. wcel 1819    =/= wne 2652   class class class wbr 4456   ` cfv 5594   RRcr 9508   0cc0 9509   1c1 9510   +oocpnf 9642    < clt 9645   NNcn 10556   NN0cn0 10816   #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-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-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-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  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-n0 10817  df-z 10886  df-uz 11107  df-hash 12409
This theorem is referenced by:  01eq0ring  18047  tgldimor  24019  frgrawopreg  25176
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