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Theorem sdom2en01 7928
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 7052 . . . . 5  |-  om  =  ( On  i^i  Fin )
2 inss2 3390 . . . . 5  |-  ( On 
i^i  Fin )  C_  Fin
31, 2eqsstri 3208 . . . 4  |-  om  C_  Fin
4 2onn 6638 . . . 4  |-  2o  e.  om
53, 4sselii 3177 . . 3  |-  2o  e.  Fin
6 sdomdom 6889 . . 3  |-  ( A 
~<  2o  ->  A  ~<_  2o )
7 domfi 7084 . . 3  |-  ( ( 2o  e.  Fin  /\  A  ~<_  2o )  ->  A  e.  Fin )
85, 6, 7sylancr 644 . 2  |-  ( A 
~<  2o  ->  A  e.  Fin )
9 id 19 . . . 4  |-  ( A  =  (/)  ->  A  =  (/) )
10 0fin 7087 . . . 4  |-  (/)  e.  Fin
119, 10syl6eqel 2371 . . 3  |-  ( A  =  (/)  ->  A  e. 
Fin )
12 1onn 6637 . . . . 5  |-  1o  e.  om
133, 12sselii 3177 . . . 4  |-  1o  e.  Fin
14 enfi 7079 . . . 4  |-  ( A 
~~  1o  ->  ( A  e.  Fin  <->  1o  e.  Fin ) )
1513, 14mpbiri 224 . . 3  |-  ( A 
~~  1o  ->  A  e. 
Fin )
1611, 15jaoi 368 . 2  |-  ( ( A  =  (/)  \/  A  ~~  1o )  ->  A  e.  Fin )
17 df2o3 6492 . . . . . 6  |-  2o  =  { (/) ,  1o }
1817eleq2i 2347 . . . . 5  |-  ( (
card `  A )  e.  2o  <->  ( card `  A
)  e.  { (/) ,  1o } )
19 fvex 5539 . . . . . 6  |-  ( card `  A )  e.  _V
2019elpr 3658 . . . . 5  |-  ( (
card `  A )  e.  { (/) ,  1o }  <->  ( ( card `  A
)  =  (/)  \/  ( card `  A )  =  1o ) )
2118, 20bitri 240 . . . 4  |-  ( (
card `  A )  e.  2o  <->  ( ( card `  A )  =  (/)  \/  ( card `  A
)  =  1o ) )
2221a1i 10 . . 3  |-  ( A  e.  Fin  ->  (
( card `  A )  e.  2o  <->  ( ( card `  A )  =  (/)  \/  ( card `  A
)  =  1o ) ) )
23 cardnn 7596 . . . . . 6  |-  ( 2o  e.  om  ->  ( card `  2o )  =  2o )
244, 23ax-mp 8 . . . . 5  |-  ( card `  2o )  =  2o
2524eleq2i 2347 . . . 4  |-  ( (
card `  A )  e.  ( card `  2o ) 
<->  ( card `  A
)  e.  2o )
26 finnum 7581 . . . . 5  |-  ( A  e.  Fin  ->  A  e.  dom  card )
27 2on 6487 . . . . . 6  |-  2o  e.  On
28 onenon 7582 . . . . . 6  |-  ( 2o  e.  On  ->  2o  e.  dom  card )
2927, 28ax-mp 8 . . . . 5  |-  2o  e.  dom  card
30 cardsdom2 7621 . . . . 5  |-  ( ( A  e.  dom  card  /\  2o  e.  dom  card )  ->  ( ( card `  A )  e.  (
card `  2o )  <->  A 
~<  2o ) )
3126, 29, 30sylancl 643 . . . 4  |-  ( A  e.  Fin  ->  (
( card `  A )  e.  ( card `  2o ) 
<->  A  ~<  2o )
)
3225, 31syl5bbr 250 . . 3  |-  ( A  e.  Fin  ->  (
( card `  A )  e.  2o  <->  A  ~<  2o ) )
33 cardnueq0 7597 . . . . 5  |-  ( A  e.  dom  card  ->  ( ( card `  A
)  =  (/)  <->  A  =  (/) ) )
3426, 33syl 15 . . . 4  |-  ( A  e.  Fin  ->  (
( card `  A )  =  (/)  <->  A  =  (/) ) )
35 cardnn 7596 . . . . . . 7  |-  ( 1o  e.  om  ->  ( card `  1o )  =  1o )
3612, 35ax-mp 8 . . . . . 6  |-  ( card `  1o )  =  1o
3736eqeq2i 2293 . . . . 5  |-  ( (
card `  A )  =  ( card `  1o ) 
<->  ( card `  A
)  =  1o )
38 finnum 7581 . . . . . . 7  |-  ( 1o  e.  Fin  ->  1o  e.  dom  card )
3913, 38ax-mp 8 . . . . . 6  |-  1o  e.  dom  card
40 carden2 7620 . . . . . 6  |-  ( ( A  e.  dom  card  /\  1o  e.  dom  card )  ->  ( ( card `  A )  =  (
card `  1o )  <->  A 
~~  1o ) )
4126, 39, 40sylancl 643 . . . . 5  |-  ( A  e.  Fin  ->  (
( card `  A )  =  ( card `  1o ) 
<->  A  ~~  1o ) )
4237, 41syl5bbr 250 . . . 4  |-  ( A  e.  Fin  ->  (
( card `  A )  =  1o  <->  A  ~~  1o ) )
4334, 42orbi12d 690 . . 3  |-  ( A  e.  Fin  ->  (
( ( card `  A
)  =  (/)  \/  ( card `  A )  =  1o )  <->  ( A  =  (/)  \/  A  ~~  1o ) ) )
4422, 32, 433bitr3d 274 . 2  |-  ( A  e.  Fin  ->  ( A  ~<  2o  <->  ( A  =  (/)  \/  A  ~~  1o ) ) )
458, 16, 44pm5.21nii 342 1  |-  ( A 
~<  2o  <->  ( A  =  (/)  \/  A  ~~  1o ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/ wo 357    = wceq 1623    e. wcel 1684    i^i cin 3151   (/)c0 3455   {cpr 3641   class class class wbr 4023   Oncon0 4392   omcom 4656   dom cdm 4689   ` cfv 5255   1oc1o 6472   2oc2o 6473    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862   Fincfn 6863   cardccrd 7568
This theorem is referenced by:  fin56  8019  en2top  16723
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-2o 6480  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572
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