MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sdom2en01 Structured version   Unicode version

Theorem sdom2en01 8182
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 7298 . . . . 5  |-  om  =  ( On  i^i  Fin )
2 inss2 3562 . . . . 5  |-  ( On 
i^i  Fin )  C_  Fin
31, 2eqsstri 3378 . . . 4  |-  om  C_  Fin
4 2onn 6883 . . . 4  |-  2o  e.  om
53, 4sselii 3345 . . 3  |-  2o  e.  Fin
6 sdomdom 7135 . . 3  |-  ( A 
~<  2o  ->  A  ~<_  2o )
7 domfi 7330 . . 3  |-  ( ( 2o  e.  Fin  /\  A  ~<_  2o )  ->  A  e.  Fin )
85, 6, 7sylancr 645 . 2  |-  ( A 
~<  2o  ->  A  e.  Fin )
9 id 20 . . . 4  |-  ( A  =  (/)  ->  A  =  (/) )
10 0fin 7336 . . . 4  |-  (/)  e.  Fin
119, 10syl6eqel 2524 . . 3  |-  ( A  =  (/)  ->  A  e. 
Fin )
12 1onn 6882 . . . . 5  |-  1o  e.  om
133, 12sselii 3345 . . . 4  |-  1o  e.  Fin
14 enfi 7325 . . . 4  |-  ( A 
~~  1o  ->  ( A  e.  Fin  <->  1o  e.  Fin ) )
1513, 14mpbiri 225 . . 3  |-  ( A 
~~  1o  ->  A  e. 
Fin )
1611, 15jaoi 369 . 2  |-  ( ( A  =  (/)  \/  A  ~~  1o )  ->  A  e.  Fin )
17 df2o3 6737 . . . . . 6  |-  2o  =  { (/) ,  1o }
1817eleq2i 2500 . . . . 5  |-  ( (
card `  A )  e.  2o  <->  ( card `  A
)  e.  { (/) ,  1o } )
19 fvex 5742 . . . . . 6  |-  ( card `  A )  e.  _V
2019elpr 3832 . . . . 5  |-  ( (
card `  A )  e.  { (/) ,  1o }  <->  ( ( card `  A
)  =  (/)  \/  ( card `  A )  =  1o ) )
2118, 20bitri 241 . . . 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 7850 . . . . . 6  |-  ( 2o  e.  om  ->  ( card `  2o )  =  2o )
244, 23ax-mp 8 . . . . 5  |-  ( card `  2o )  =  2o
2524eleq2i 2500 . . . 4  |-  ( (
card `  A )  e.  ( card `  2o ) 
<->  ( card `  A
)  e.  2o )
26 finnum 7835 . . . . 5  |-  ( A  e.  Fin  ->  A  e.  dom  card )
27 2on 6732 . . . . . 6  |-  2o  e.  On
28 onenon 7836 . . . . . 6  |-  ( 2o  e.  On  ->  2o  e.  dom  card )
2927, 28ax-mp 8 . . . . 5  |-  2o  e.  dom  card
30 cardsdom2 7875 . . . . 5  |-  ( ( A  e.  dom  card  /\  2o  e.  dom  card )  ->  ( ( card `  A )  e.  (
card `  2o )  <->  A 
~<  2o ) )
3126, 29, 30sylancl 644 . . . 4  |-  ( A  e.  Fin  ->  (
( card `  A )  e.  ( card `  2o ) 
<->  A  ~<  2o )
)
3225, 31syl5bbr 251 . . 3  |-  ( A  e.  Fin  ->  (
( card `  A )  e.  2o  <->  A  ~<  2o ) )
33 cardnueq0 7851 . . . . 5  |-  ( A  e.  dom  card  ->  ( ( card `  A
)  =  (/)  <->  A  =  (/) ) )
3426, 33syl 16 . . . 4  |-  ( A  e.  Fin  ->  (
( card `  A )  =  (/)  <->  A  =  (/) ) )
35 cardnn 7850 . . . . . . 7  |-  ( 1o  e.  om  ->  ( card `  1o )  =  1o )
3612, 35ax-mp 8 . . . . . 6  |-  ( card `  1o )  =  1o
3736eqeq2i 2446 . . . . 5  |-  ( (
card `  A )  =  ( card `  1o ) 
<->  ( card `  A
)  =  1o )
38 finnum 7835 . . . . . . 7  |-  ( 1o  e.  Fin  ->  1o  e.  dom  card )
3913, 38ax-mp 8 . . . . . 6  |-  1o  e.  dom  card
40 carden2 7874 . . . . . 6  |-  ( ( A  e.  dom  card  /\  1o  e.  dom  card )  ->  ( ( card `  A )  =  (
card `  1o )  <->  A 
~~  1o ) )
4126, 39, 40sylancl 644 . . . . 5  |-  ( A  e.  Fin  ->  (
( card `  A )  =  ( card `  1o ) 
<->  A  ~~  1o ) )
4237, 41syl5bbr 251 . . . 4  |-  ( A  e.  Fin  ->  (
( card `  A )  =  1o  <->  A  ~~  1o ) )
4334, 42orbi12d 691 . . 3  |-  ( A  e.  Fin  ->  (
( ( card `  A
)  =  (/)  \/  ( card `  A )  =  1o )  <->  ( A  =  (/)  \/  A  ~~  1o ) ) )
4422, 32, 433bitr3d 275 . 2  |-  ( A  e.  Fin  ->  ( A  ~<  2o  <->  ( A  =  (/)  \/  A  ~~  1o ) ) )
458, 16, 44pm5.21nii 343 1  |-  ( A 
~<  2o  <->  ( A  =  (/)  \/  A  ~~  1o ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    \/ wo 358    = wceq 1652    e. wcel 1725    i^i cin 3319   (/)c0 3628   {cpr 3815   class class class wbr 4212   Oncon0 4581   omcom 4845   dom cdm 4878   ` cfv 5454   1oc1o 6717   2oc2o 6718    ~~ cen 7106    ~<_ cdom 7107    ~< csdm 7108   Fincfn 7109   cardccrd 7822
This theorem is referenced by:  fin56  8273  en2top  17050
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-1o 6724  df-2o 6725  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826
  Copyright terms: Public domain W3C validator