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Theorem sdom2en01 8699
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 7728 . . . . 5  |-  om  =  ( On  i^i  Fin )
2 inss2 3715 . . . . 5  |-  ( On 
i^i  Fin )  C_  Fin
31, 2eqsstri 3529 . . . 4  |-  om  C_  Fin
4 2onn 7307 . . . 4  |-  2o  e.  om
53, 4sselii 3496 . . 3  |-  2o  e.  Fin
6 sdomdom 7562 . . 3  |-  ( A 
~<  2o  ->  A  ~<_  2o )
7 domfi 7760 . . 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 7766 . . . 4  |-  (/)  e.  Fin
119, 10syl6eqel 2553 . . 3  |-  ( A  =  (/)  ->  A  e. 
Fin )
12 1onn 7306 . . . . 5  |-  1o  e.  om
133, 12sselii 3496 . . . 4  |-  1o  e.  Fin
14 enfi 7755 . . . 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 7161 . . . . . 6  |-  2o  =  { (/) ,  1o }
1817eleq2i 2535 . . . . 5  |-  ( (
card `  A )  e.  2o  <->  ( card `  A
)  e.  { (/) ,  1o } )
19 fvex 5882 . . . . . 6  |-  ( card `  A )  e.  _V
2019elpr 4050 . . . . 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 8361 . . . . . 6  |-  ( 2o  e.  om  ->  ( card `  2o )  =  2o )
244, 23ax-mp 5 . . . . 5  |-  ( card `  2o )  =  2o
2524eleq2i 2535 . . . 4  |-  ( (
card `  A )  e.  ( card `  2o ) 
<->  ( card `  A
)  e.  2o )
26 finnum 8346 . . . . 5  |-  ( A  e.  Fin  ->  A  e.  dom  card )
27 2on 7156 . . . . . 6  |-  2o  e.  On
28 onenon 8347 . . . . . 6  |-  ( 2o  e.  On  ->  2o  e.  dom  card )
2927, 28ax-mp 5 . . . . 5  |-  2o  e.  dom  card
30 cardsdom2 8386 . . . . 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 8362 . . . . 5  |-  ( A  e.  dom  card  ->  ( ( card `  A
)  =  (/)  <->  A  =  (/) ) )
3426, 33syl 16 . . . 4  |-  ( A  e.  Fin  ->  (
( card `  A )  =  (/)  <->  A  =  (/) ) )
35 cardnn 8361 . . . . . . 7  |-  ( 1o  e.  om  ->  ( card `  1o )  =  1o )
3612, 35ax-mp 5 . . . . . 6  |-  ( card `  1o )  =  1o
3736eqeq2i 2475 . . . . 5  |-  ( (
card `  A )  =  ( card `  1o ) 
<->  ( card `  A
)  =  1o )
38 finnum 8346 . . . . . . 7  |-  ( 1o  e.  Fin  ->  1o  e.  dom  card )
3913, 38ax-mp 5 . . . . . 6  |-  1o  e.  dom  card
40 carden2 8385 . . . . . 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 1395    e. wcel 1819    i^i cin 3470   (/)c0 3793   {cpr 4034   class class class wbr 4456   Oncon0 4887   dom cdm 5008   ` cfv 5594   omcom 6699   1oc1o 7141   2oc2o 7142    ~~ cen 7532    ~<_ cdom 7533    ~< csdm 7534   Fincfn 7535   cardccrd 8333
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
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-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  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-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-om 6700  df-1o 7148  df-2o 7149  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337
This theorem is referenced by:  fin56  8790  en2top  19613
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