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

Theorem cardne 7685
Description: No member of a cardinal number of a set is equinumerous to the set. Proposition 10.6(2) of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 9-Jan-2013.)
Assertion
Ref Expression
cardne  |-  ( A  e.  ( card `  B
)  ->  -.  A  ~~  B )

Proof of Theorem cardne
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elfvdm 5634 . 2  |-  ( A  e.  ( card `  B
)  ->  B  e.  dom  card )
2 cardon 7664 . . . . . . . . . 10  |-  ( card `  B )  e.  On
32oneli 4579 . . . . . . . . 9  |-  ( A  e.  ( card `  B
)  ->  A  e.  On )
4 breq1 4105 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  ~~  B  <->  A  ~~  B ) )
54onintss 4521 . . . . . . . . 9  |-  ( A  e.  On  ->  ( A  ~~  B  ->  |^| { x  e.  On  |  x  ~~  B }  C_  A ) )
63, 5syl 15 . . . . . . . 8  |-  ( A  e.  ( card `  B
)  ->  ( A  ~~  B  ->  |^| { x  e.  On  |  x  ~~  B }  C_  A ) )
76adantl 452 . . . . . . 7  |-  ( ( B  e.  dom  card  /\  A  e.  ( card `  B ) )  -> 
( A  ~~  B  ->  |^| { x  e.  On  |  x  ~~  B }  C_  A ) )
8 cardval3 7672 . . . . . . . . 9  |-  ( B  e.  dom  card  ->  (
card `  B )  =  |^| { x  e.  On  |  x  ~~  B } )
98sseq1d 3281 . . . . . . . 8  |-  ( B  e.  dom  card  ->  ( ( card `  B
)  C_  A  <->  |^| { x  e.  On  |  x  ~~  B }  C_  A ) )
109adantr 451 . . . . . . 7  |-  ( ( B  e.  dom  card  /\  A  e.  ( card `  B ) )  -> 
( ( card `  B
)  C_  A  <->  |^| { x  e.  On  |  x  ~~  B }  C_  A ) )
117, 10sylibrd 225 . . . . . 6  |-  ( ( B  e.  dom  card  /\  A  e.  ( card `  B ) )  -> 
( A  ~~  B  ->  ( card `  B
)  C_  A )
)
12 ontri1 4505 . . . . . . . 8  |-  ( ( ( card `  B
)  e.  On  /\  A  e.  On )  ->  ( ( card `  B
)  C_  A  <->  -.  A  e.  ( card `  B
) ) )
132, 3, 12sylancr 644 . . . . . . 7  |-  ( A  e.  ( card `  B
)  ->  ( ( card `  B )  C_  A 
<->  -.  A  e.  (
card `  B )
) )
1413adantl 452 . . . . . 6  |-  ( ( B  e.  dom  card  /\  A  e.  ( card `  B ) )  -> 
( ( card `  B
)  C_  A  <->  -.  A  e.  ( card `  B
) ) )
1511, 14sylibd 205 . . . . 5  |-  ( ( B  e.  dom  card  /\  A  e.  ( card `  B ) )  -> 
( A  ~~  B  ->  -.  A  e.  (
card `  B )
) )
1615con2d 107 . . . 4  |-  ( ( B  e.  dom  card  /\  A  e.  ( card `  B ) )  -> 
( A  e.  (
card `  B )  ->  -.  A  ~~  B
) )
1716ex 423 . . 3  |-  ( B  e.  dom  card  ->  ( A  e.  ( card `  B )  ->  ( A  e.  ( card `  B )  ->  -.  A  ~~  B ) ) )
1817pm2.43d 44 . 2  |-  ( B  e.  dom  card  ->  ( A  e.  ( card `  B )  ->  -.  A  ~~  B ) )
191, 18mpcom 32 1  |-  ( A  e.  ( card `  B
)  ->  -.  A  ~~  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1710   {crab 2623    C_ wss 3228   |^|cint 3941   class class class wbr 4102   Oncon0 4471   dom cdm 4768   ` cfv 5334    ~~ cen 6945   cardccrd 7655
This theorem is referenced by:  carden2b  7687  cardlim  7692  cardsdomelir  7693
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-fv 5342  df-en 6949  df-card 7659
  Copyright terms: Public domain W3C validator