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Theorem cardne 7598
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 5554 . 2  |-  ( A  e.  ( card `  B
)  ->  B  e.  dom  card )
2 cardon 7577 . . . . . . . . . 10  |-  ( card `  B )  e.  On
32oneli 4500 . . . . . . . . 9  |-  ( A  e.  ( card `  B
)  ->  A  e.  On )
4 breq1 4026 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  ~~  B  <->  A  ~~  B ) )
54onintss 4442 . . . . . . . . 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 7585 . . . . . . . . 9  |-  ( B  e.  dom  card  ->  (
card `  B )  =  |^| { x  e.  On  |  x  ~~  B } )
98sseq1d 3205 . . . . . . . 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 4426 . . . . . . . 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 1684   {crab 2547    C_ wss 3152   |^|cint 3862   class class class wbr 4023   Oncon0 4392   dom cdm 4689   ` cfv 5255    ~~ cen 6860   cardccrd 7568
This theorem is referenced by:  carden2b  7600  cardlim  7605  cardsdomelir  7606
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-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-fv 5263  df-en 6864  df-card 7572
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