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Theorem cardne 8363
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 5898 . 2  |-  ( A  e.  ( card `  B
)  ->  B  e.  dom  card )
2 cardon 8342 . . . . . . . . . 10  |-  ( card `  B )  e.  On
32oneli 4994 . . . . . . . . 9  |-  ( A  e.  ( card `  B
)  ->  A  e.  On )
4 breq1 4459 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  ~~  B  <->  A  ~~  B ) )
54onintss 4937 . . . . . . . . 9  |-  ( A  e.  On  ->  ( A  ~~  B  ->  |^| { x  e.  On  |  x  ~~  B }  C_  A ) )
63, 5syl 16 . . . . . . . 8  |-  ( A  e.  ( card `  B
)  ->  ( A  ~~  B  ->  |^| { x  e.  On  |  x  ~~  B }  C_  A ) )
76adantl 466 . . . . . . 7  |-  ( ( B  e.  dom  card  /\  A  e.  ( card `  B ) )  -> 
( A  ~~  B  ->  |^| { x  e.  On  |  x  ~~  B }  C_  A ) )
8 cardval3 8350 . . . . . . . . 9  |-  ( B  e.  dom  card  ->  (
card `  B )  =  |^| { x  e.  On  |  x  ~~  B } )
98sseq1d 3526 . . . . . . . 8  |-  ( B  e.  dom  card  ->  ( ( card `  B
)  C_  A  <->  |^| { x  e.  On  |  x  ~~  B }  C_  A ) )
109adantr 465 . . . . . . 7  |-  ( ( B  e.  dom  card  /\  A  e.  ( card `  B ) )  -> 
( ( card `  B
)  C_  A  <->  |^| { x  e.  On  |  x  ~~  B }  C_  A ) )
117, 10sylibrd 234 . . . . . 6  |-  ( ( B  e.  dom  card  /\  A  e.  ( card `  B ) )  -> 
( A  ~~  B  ->  ( card `  B
)  C_  A )
)
12 ontri1 4921 . . . . . . . 8  |-  ( ( ( card `  B
)  e.  On  /\  A  e.  On )  ->  ( ( card `  B
)  C_  A  <->  -.  A  e.  ( card `  B
) ) )
132, 3, 12sylancr 663 . . . . . . 7  |-  ( A  e.  ( card `  B
)  ->  ( ( card `  B )  C_  A 
<->  -.  A  e.  (
card `  B )
) )
1413adantl 466 . . . . . 6  |-  ( ( B  e.  dom  card  /\  A  e.  ( card `  B ) )  -> 
( ( card `  B
)  C_  A  <->  -.  A  e.  ( card `  B
) ) )
1511, 14sylibd 214 . . . . 5  |-  ( ( B  e.  dom  card  /\  A  e.  ( card `  B ) )  -> 
( A  ~~  B  ->  -.  A  e.  (
card `  B )
) )
1615con2d 115 . . . 4  |-  ( ( B  e.  dom  card  /\  A  e.  ( card `  B ) )  -> 
( A  e.  (
card `  B )  ->  -.  A  ~~  B
) )
1716ex 434 . . 3  |-  ( B  e.  dom  card  ->  ( A  e.  ( card `  B )  ->  ( A  e.  ( card `  B )  ->  -.  A  ~~  B ) ) )
1817pm2.43d 48 . 2  |-  ( B  e.  dom  card  ->  ( A  e.  ( card `  B )  ->  -.  A  ~~  B ) )
191, 18mpcom 36 1  |-  ( A  e.  ( card `  B
)  ->  -.  A  ~~  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1819   {crab 2811    C_ wss 3471   |^|cint 4288   class class class wbr 4456   Oncon0 4887   dom cdm 5008   ` cfv 5594    ~~ cen 7532   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-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-fv 5602  df-en 7536  df-card 8337
This theorem is referenced by:  carden2b  8365  cardlim  8370  cardsdomelir  8371
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