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Theorem cardne 7593
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 )
Dummy variable  x is distinct from all other variables.

Proof of Theorem cardne
StepHypRef Expression
1 elfvdm 5515 . 2  |-  ( A  e.  ( card `  B
)  ->  B  e.  dom  card )
2 cardon 7572 . . . . . . . . . 10  |-  ( card `  B )  e.  On
32oneli 4499 . . . . . . . . 9  |-  ( A  e.  ( card `  B
)  ->  A  e.  On )
4 breq1 4027 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  ~~  B  <->  A  ~~  B ) )
54onintss 4441 . . . . . . . . 9  |-  ( A  e.  On  ->  ( A  ~~  B  ->  |^| { x  e.  On  |  x  ~~  B }  C_  A ) )
63, 5syl 17 . . . . . . . 8  |-  ( A  e.  ( card `  B
)  ->  ( A  ~~  B  ->  |^| { x  e.  On  |  x  ~~  B }  C_  A ) )
76adantl 454 . . . . . . 7  |-  ( ( B  e.  dom  card  /\  A  e.  ( card `  B ) )  -> 
( A  ~~  B  ->  |^| { x  e.  On  |  x  ~~  B }  C_  A ) )
8 cardval3 7580 . . . . . . . . 9  |-  ( B  e.  dom  card  ->  (
card `  B )  =  |^| { x  e.  On  |  x  ~~  B } )
98sseq1d 3206 . . . . . . . 8  |-  ( B  e.  dom  card  ->  ( ( card `  B
)  C_  A  <->  |^| { x  e.  On  |  x  ~~  B }  C_  A ) )
109adantr 453 . . . . . . 7  |-  ( ( B  e.  dom  card  /\  A  e.  ( card `  B ) )  -> 
( ( card `  B
)  C_  A  <->  |^| { x  e.  On  |  x  ~~  B }  C_  A ) )
117, 10sylibrd 227 . . . . . 6  |-  ( ( B  e.  dom  card  /\  A  e.  ( card `  B ) )  -> 
( A  ~~  B  ->  ( card `  B
)  C_  A )
)
12 ontri1 4425 . . . . . . . 8  |-  ( ( ( card `  B
)  e.  On  /\  A  e.  On )  ->  ( ( card `  B
)  C_  A  <->  -.  A  e.  ( card `  B
) ) )
132, 3, 12sylancr 646 . . . . . . 7  |-  ( A  e.  ( card `  B
)  ->  ( ( card `  B )  C_  A 
<->  -.  A  e.  (
card `  B )
) )
1413adantl 454 . . . . . 6  |-  ( ( B  e.  dom  card  /\  A  e.  ( card `  B ) )  -> 
( ( card `  B
)  C_  A  <->  -.  A  e.  ( card `  B
) ) )
1511, 14sylibd 207 . . . . 5  |-  ( ( B  e.  dom  card  /\  A  e.  ( card `  B ) )  -> 
( A  ~~  B  ->  -.  A  e.  (
card `  B )
) )
1615con2d 109 . . . 4  |-  ( ( B  e.  dom  card  /\  A  e.  ( card `  B ) )  -> 
( A  e.  (
card `  B )  ->  -.  A  ~~  B
) )
1716ex 425 . . 3  |-  ( B  e.  dom  card  ->  ( A  e.  ( card `  B )  ->  ( A  e.  ( card `  B )  ->  -.  A  ~~  B ) ) )
1817pm2.43d 46 . 2  |-  ( B  e.  dom  card  ->  ( A  e.  ( card `  B )  ->  -.  A  ~~  B ) )
191, 18mpcom 34 1  |-  ( A  e.  ( card `  B
)  ->  -.  A  ~~  B )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    e. wcel 1685   {crab 2548    C_ wss 3153   |^|cint 3863   class class class wbr 4024   Oncon0 4391   dom cdm 4688   ` cfv 5221    ~~ cen 6855   cardccrd 7563
This theorem is referenced by:  carden2b  7595  cardlim  7600  cardsdomelir  7601
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-fv 5229  df-en 6859  df-card 7567
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