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Theorem sdom2en01 8681
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 7709 . . . . 5  |-  om  =  ( On  i^i  Fin )
2 inss2 3719 . . . . 5  |-  ( On 
i^i  Fin )  C_  Fin
31, 2eqsstri 3534 . . . 4  |-  om  C_  Fin
4 2onn 7289 . . . 4  |-  2o  e.  om
53, 4sselii 3501 . . 3  |-  2o  e.  Fin
6 sdomdom 7543 . . 3  |-  ( A 
~<  2o  ->  A  ~<_  2o )
7 domfi 7741 . . 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 7747 . . . 4  |-  (/)  e.  Fin
119, 10syl6eqel 2563 . . 3  |-  ( A  =  (/)  ->  A  e. 
Fin )
12 1onn 7288 . . . . 5  |-  1o  e.  om
133, 12sselii 3501 . . . 4  |-  1o  e.  Fin
14 enfi 7736 . . . 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 7143 . . . . . 6  |-  2o  =  { (/) ,  1o }
1817eleq2i 2545 . . . . 5  |-  ( (
card `  A )  e.  2o  <->  ( card `  A
)  e.  { (/) ,  1o } )
19 fvex 5875 . . . . . 6  |-  ( card `  A )  e.  _V
2019elpr 4045 . . . . 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 8343 . . . . . 6  |-  ( 2o  e.  om  ->  ( card `  2o )  =  2o )
244, 23ax-mp 5 . . . . 5  |-  ( card `  2o )  =  2o
2524eleq2i 2545 . . . 4  |-  ( (
card `  A )  e.  ( card `  2o ) 
<->  ( card `  A
)  e.  2o )
26 finnum 8328 . . . . 5  |-  ( A  e.  Fin  ->  A  e.  dom  card )
27 2on 7138 . . . . . 6  |-  2o  e.  On
28 onenon 8329 . . . . . 6  |-  ( 2o  e.  On  ->  2o  e.  dom  card )
2927, 28ax-mp 5 . . . . 5  |-  2o  e.  dom  card
30 cardsdom2 8368 . . . . 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 8344 . . . . 5  |-  ( A  e.  dom  card  ->  ( ( card `  A
)  =  (/)  <->  A  =  (/) ) )
3426, 33syl 16 . . . 4  |-  ( A  e.  Fin  ->  (
( card `  A )  =  (/)  <->  A  =  (/) ) )
35 cardnn 8343 . . . . . . 7  |-  ( 1o  e.  om  ->  ( card `  1o )  =  1o )
3612, 35ax-mp 5 . . . . . 6  |-  ( card `  1o )  =  1o
3736eqeq2i 2485 . . . . 5  |-  ( (
card `  A )  =  ( card `  1o ) 
<->  ( card `  A
)  =  1o )
38 finnum 8328 . . . . . . 7  |-  ( 1o  e.  Fin  ->  1o  e.  dom  card )
3913, 38ax-mp 5 . . . . . 6  |-  1o  e.  dom  card
40 carden2 8367 . . . . . 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 1379    e. wcel 1767    i^i cin 3475   (/)c0 3785   {cpr 4029   class class class wbr 4447   Oncon0 4878   dom cdm 4999   ` cfv 5587   omcom 6679   1oc1o 7123   2oc2o 7124    ~~ cen 7513    ~<_ cdom 7514    ~< csdm 7515   Fincfn 7516   cardccrd 8315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-om 6680  df-1o 7130  df-2o 7131  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8319
This theorem is referenced by:  fin56  8772  en2top  19269
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