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Theorem sdom2en01 8483
Description: A set with less than two elements has 0 or 1. (Contributed by Stefan O'Rear, 30-Oct-2014.)
Assertion
Ref Expression
sdom2en01  |-  ( A 
~<  2o  <->  ( A  =  (/)  \/  A  ~~  1o ) )

Proof of Theorem sdom2en01
StepHypRef Expression
1 onfin2 7514 . . . . 5  |-  om  =  ( On  i^i  Fin )
2 inss2 3583 . . . . 5  |-  ( On 
i^i  Fin )  C_  Fin
31, 2eqsstri 3398 . . . 4  |-  om  C_  Fin
4 2onn 7091 . . . 4  |-  2o  e.  om
53, 4sselii 3365 . . 3  |-  2o  e.  Fin
6 sdomdom 7349 . . 3  |-  ( A 
~<  2o  ->  A  ~<_  2o )
7 domfi 7546 . . 3  |-  ( ( 2o  e.  Fin  /\  A  ~<_  2o )  ->  A  e.  Fin )
85, 6, 7sylancr 663 . 2  |-  ( A 
~<  2o  ->  A  e.  Fin )
9 id 22 . . . 4  |-  ( A  =  (/)  ->  A  =  (/) )
10 0fin 7552 . . . 4  |-  (/)  e.  Fin
119, 10syl6eqel 2531 . . 3  |-  ( A  =  (/)  ->  A  e. 
Fin )
12 1onn 7090 . . . . 5  |-  1o  e.  om
133, 12sselii 3365 . . . 4  |-  1o  e.  Fin
14 enfi 7541 . . . 4  |-  ( A 
~~  1o  ->  ( A  e.  Fin  <->  1o  e.  Fin ) )
1513, 14mpbiri 233 . . 3  |-  ( A 
~~  1o  ->  A  e. 
Fin )
1611, 15jaoi 379 . 2  |-  ( ( A  =  (/)  \/  A  ~~  1o )  ->  A  e.  Fin )
17 df2o3 6945 . . . . . 6  |-  2o  =  { (/) ,  1o }
1817eleq2i 2507 . . . . 5  |-  ( (
card `  A )  e.  2o  <->  ( card `  A
)  e.  { (/) ,  1o } )
19 fvex 5713 . . . . . 6  |-  ( card `  A )  e.  _V
2019elpr 3907 . . . . 5  |-  ( (
card `  A )  e.  { (/) ,  1o }  <->  ( ( card `  A
)  =  (/)  \/  ( card `  A )  =  1o ) )
2118, 20bitri 249 . . . 4  |-  ( (
card `  A )  e.  2o  <->  ( ( card `  A )  =  (/)  \/  ( card `  A
)  =  1o ) )
2221a1i 11 . . 3  |-  ( A  e.  Fin  ->  (
( card `  A )  e.  2o  <->  ( ( card `  A )  =  (/)  \/  ( card `  A
)  =  1o ) ) )
23 cardnn 8145 . . . . . 6  |-  ( 2o  e.  om  ->  ( card `  2o )  =  2o )
244, 23ax-mp 5 . . . . 5  |-  ( card `  2o )  =  2o
2524eleq2i 2507 . . . 4  |-  ( (
card `  A )  e.  ( card `  2o ) 
<->  ( card `  A
)  e.  2o )
26 finnum 8130 . . . . 5  |-  ( A  e.  Fin  ->  A  e.  dom  card )
27 2on 6940 . . . . . 6  |-  2o  e.  On
28 onenon 8131 . . . . . 6  |-  ( 2o  e.  On  ->  2o  e.  dom  card )
2927, 28ax-mp 5 . . . . 5  |-  2o  e.  dom  card
30 cardsdom2 8170 . . . . 5  |-  ( ( A  e.  dom  card  /\  2o  e.  dom  card )  ->  ( ( card `  A )  e.  (
card `  2o )  <->  A 
~<  2o ) )
3126, 29, 30sylancl 662 . . . 4  |-  ( A  e.  Fin  ->  (
( card `  A )  e.  ( card `  2o ) 
<->  A  ~<  2o )
)
3225, 31syl5bbr 259 . . 3  |-  ( A  e.  Fin  ->  (
( card `  A )  e.  2o  <->  A  ~<  2o ) )
33 cardnueq0 8146 . . . . 5  |-  ( A  e.  dom  card  ->  ( ( card `  A
)  =  (/)  <->  A  =  (/) ) )
3426, 33syl 16 . . . 4  |-  ( A  e.  Fin  ->  (
( card `  A )  =  (/)  <->  A  =  (/) ) )
35 cardnn 8145 . . . . . . 7  |-  ( 1o  e.  om  ->  ( card `  1o )  =  1o )
3612, 35ax-mp 5 . . . . . 6  |-  ( card `  1o )  =  1o
3736eqeq2i 2453 . . . . 5  |-  ( (
card `  A )  =  ( card `  1o ) 
<->  ( card `  A
)  =  1o )
38 finnum 8130 . . . . . . 7  |-  ( 1o  e.  Fin  ->  1o  e.  dom  card )
3913, 38ax-mp 5 . . . . . 6  |-  1o  e.  dom  card
40 carden2 8169 . . . . . 6  |-  ( ( A  e.  dom  card  /\  1o  e.  dom  card )  ->  ( ( card `  A )  =  (
card `  1o )  <->  A 
~~  1o ) )
4126, 39, 40sylancl 662 . . . . 5  |-  ( A  e.  Fin  ->  (
( card `  A )  =  ( card `  1o ) 
<->  A  ~~  1o ) )
4237, 41syl5bbr 259 . . . 4  |-  ( A  e.  Fin  ->  (
( card `  A )  =  1o  <->  A  ~~  1o ) )
4334, 42orbi12d 709 . . 3  |-  ( A  e.  Fin  ->  (
( ( card `  A
)  =  (/)  \/  ( card `  A )  =  1o )  <->  ( A  =  (/)  \/  A  ~~  1o ) ) )
4422, 32, 433bitr3d 283 . 2  |-  ( A  e.  Fin  ->  ( A  ~<  2o  <->  ( A  =  (/)  \/  A  ~~  1o ) ) )
458, 16, 44pm5.21nii 353 1  |-  ( A 
~<  2o  <->  ( A  =  (/)  \/  A  ~~  1o ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 368    = wceq 1369    e. wcel 1756    i^i cin 3339   (/)c0 3649   {cpr 3891   class class class wbr 4304   Oncon0 4731   dom cdm 4852   ` cfv 5430   omcom 6488   1oc1o 6925   2oc2o 6926    ~~ cen 7319    ~<_ cdom 7320    ~< csdm 7321   Fincfn 7322   cardccrd 8117
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 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-om 6489  df-1o 6932  df-2o 6933  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-card 8121
This theorem is referenced by:  fin56  8574  en2top  18602
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