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Theorem carden2a 7809
Description: If two sets have equal nonzero cardinalities, then they are equinumerous. (This assertion and carden2b 7810 are meant to replace carden 8382 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.)
Assertion
Ref Expression
carden2a  |-  ( ( ( card `  A
)  =  ( card `  B )  /\  ( card `  A )  =/=  (/) )  ->  A  ~~  B )

Proof of Theorem carden2a
StepHypRef Expression
1 df-ne 2569 . 2  |-  ( (
card `  A )  =/=  (/)  <->  -.  ( card `  A )  =  (/) )
2 ndmfv 5714 . . . . . . 7  |-  ( -.  B  e.  dom  card  -> 
( card `  B )  =  (/) )
3 eqeq1 2410 . . . . . . 7  |-  ( (
card `  A )  =  ( card `  B
)  ->  ( ( card `  A )  =  (/) 
<->  ( card `  B
)  =  (/) ) )
42, 3syl5ibr 213 . . . . . 6  |-  ( (
card `  A )  =  ( card `  B
)  ->  ( -.  B  e.  dom  card  ->  (
card `  A )  =  (/) ) )
54con1d 118 . . . . 5  |-  ( (
card `  A )  =  ( card `  B
)  ->  ( -.  ( card `  A )  =  (/)  ->  B  e.  dom  card ) )
65imp 419 . . . 4  |-  ( ( ( card `  A
)  =  ( card `  B )  /\  -.  ( card `  A )  =  (/) )  ->  B  e.  dom  card )
7 cardid2 7796 . . . 4  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
86, 7syl 16 . . 3  |-  ( ( ( card `  A
)  =  ( card `  B )  /\  -.  ( card `  A )  =  (/) )  ->  ( card `  B )  ~~  B )
9 cardid2 7796 . . . . . . 7  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
10 ndmfv 5714 . . . . . . 7  |-  ( -.  A  e.  dom  card  -> 
( card `  A )  =  (/) )
119, 10nsyl4 136 . . . . . 6  |-  ( -.  ( card `  A
)  =  (/)  ->  ( card `  A )  ~~  A )
1211ensymd 7117 . . . . 5  |-  ( -.  ( card `  A
)  =  (/)  ->  A  ~~  ( card `  A
) )
13 breq2 4176 . . . . . 6  |-  ( (
card `  A )  =  ( card `  B
)  ->  ( A  ~~  ( card `  A
)  <->  A  ~~  ( card `  B ) ) )
14 entr 7118 . . . . . . 7  |-  ( ( A  ~~  ( card `  B )  /\  ( card `  B )  ~~  B )  ->  A  ~~  B )
1514ex 424 . . . . . 6  |-  ( A 
~~  ( card `  B
)  ->  ( ( card `  B )  ~~  B  ->  A  ~~  B
) )
1613, 15syl6bi 220 . . . . 5  |-  ( (
card `  A )  =  ( card `  B
)  ->  ( A  ~~  ( card `  A
)  ->  ( ( card `  B )  ~~  B  ->  A  ~~  B
) ) )
1712, 16syl5 30 . . . 4  |-  ( (
card `  A )  =  ( card `  B
)  ->  ( -.  ( card `  A )  =  (/)  ->  ( ( card `  B )  ~~  B  ->  A  ~~  B
) ) )
1817imp 419 . . 3  |-  ( ( ( card `  A
)  =  ( card `  B )  /\  -.  ( card `  A )  =  (/) )  ->  (
( card `  B )  ~~  B  ->  A  ~~  B ) )
198, 18mpd 15 . 2  |-  ( ( ( card `  A
)  =  ( card `  B )  /\  -.  ( card `  A )  =  (/) )  ->  A  ~~  B )
201, 19sylan2b 462 1  |-  ( ( ( card `  A
)  =  ( card `  B )  /\  ( card `  A )  =/=  (/) )  ->  A  ~~  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   (/)c0 3588   class class class wbr 4172   dom cdm 4837   ` cfv 5413    ~~ cen 7065   cardccrd 7778
This theorem is referenced by:  card1  7811
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-er 6864  df-en 7069  df-card 7782
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