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Theorem carden2a 7599
Description: If two sets have equal nonzero cardinalities, then they are equinumerous. (This assertion and carden2b 7600 are meant to replace carden 8173 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 2448 . 2  |-  ( (
card `  A )  =/=  (/)  <->  -.  ( card `  A )  =  (/) )
2 ndmfv 5552 . . . . . . 7  |-  ( -.  B  e.  dom  card  -> 
( card `  B )  =  (/) )
3 eqeq1 2289 . . . . . . 7  |-  ( (
card `  A )  =  ( card `  B
)  ->  ( ( card `  A )  =  (/) 
<->  ( card `  B
)  =  (/) ) )
42, 3syl5ibr 212 . . . . . 6  |-  ( (
card `  A )  =  ( card `  B
)  ->  ( -.  B  e.  dom  card  ->  (
card `  A )  =  (/) ) )
54con1d 116 . . . . 5  |-  ( (
card `  A )  =  ( card `  B
)  ->  ( -.  ( card `  A )  =  (/)  ->  B  e.  dom  card ) )
65imp 418 . . . 4  |-  ( ( ( card `  A
)  =  ( card `  B )  /\  -.  ( card `  A )  =  (/) )  ->  B  e.  dom  card )
7 cardid2 7586 . . . 4  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
86, 7syl 15 . . 3  |-  ( ( ( card `  A
)  =  ( card `  B )  /\  -.  ( card `  A )  =  (/) )  ->  ( card `  B )  ~~  B )
9 cardid2 7586 . . . . . . 7  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
10 ndmfv 5552 . . . . . . 7  |-  ( -.  A  e.  dom  card  -> 
( card `  A )  =  (/) )
119, 10nsyl4 134 . . . . . 6  |-  ( -.  ( card `  A
)  =  (/)  ->  ( card `  A )  ~~  A )
12 ensym 6910 . . . . . 6  |-  ( (
card `  A )  ~~  A  ->  A  ~~  ( card `  A )
)
1311, 12syl 15 . . . . 5  |-  ( -.  ( card `  A
)  =  (/)  ->  A  ~~  ( card `  A
) )
14 breq2 4027 . . . . . 6  |-  ( (
card `  A )  =  ( card `  B
)  ->  ( A  ~~  ( card `  A
)  <->  A  ~~  ( card `  B ) ) )
15 entr 6913 . . . . . . 7  |-  ( ( A  ~~  ( card `  B )  /\  ( card `  B )  ~~  B )  ->  A  ~~  B )
1615ex 423 . . . . . 6  |-  ( A 
~~  ( card `  B
)  ->  ( ( card `  B )  ~~  B  ->  A  ~~  B
) )
1714, 16syl6bi 219 . . . . 5  |-  ( (
card `  A )  =  ( card `  B
)  ->  ( A  ~~  ( card `  A
)  ->  ( ( card `  B )  ~~  B  ->  A  ~~  B
) ) )
1813, 17syl5 28 . . . 4  |-  ( (
card `  A )  =  ( card `  B
)  ->  ( -.  ( card `  A )  =  (/)  ->  ( ( card `  B )  ~~  B  ->  A  ~~  B
) ) )
1918imp 418 . . 3  |-  ( ( ( card `  A
)  =  ( card `  B )  /\  -.  ( card `  A )  =  (/) )  ->  (
( card `  B )  ~~  B  ->  A  ~~  B ) )
208, 19mpd 14 . 2  |-  ( ( ( card `  A
)  =  ( card `  B )  /\  -.  ( card `  A )  =  (/) )  ->  A  ~~  B )
211, 20sylan2b 461 1  |-  ( ( ( card `  A
)  =  ( card `  B )  /\  ( card `  A )  =/=  (/) )  ->  A  ~~  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   (/)c0 3455   class class class wbr 4023   dom cdm 4689   ` cfv 5255    ~~ cen 6860   cardccrd 7568
This theorem is referenced by:  card1  7601
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-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-er 6660  df-en 6864  df-card 7572
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