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Theorem carden2a 7813
Description: If two sets have equal nonzero cardinalities, then they are equinumerous. (This assertion and carden2b 7814 are meant to replace carden 8386 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 2573 . 2  |-  ( (
card `  A )  =/=  (/)  <->  -.  ( card `  A )  =  (/) )
2 ndmfv 5718 . . . . . . 7  |-  ( -.  B  e.  dom  card  -> 
( card `  B )  =  (/) )
3 eqeq1 2414 . . . . . . 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 7800 . . . 4  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
86, 7syl 16 . . 3  |-  ( ( ( card `  A
)  =  ( card `  B )  /\  -.  ( card `  A )  =  (/) )  ->  ( card `  B )  ~~  B )
9 cardid2 7800 . . . . . . 7  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
10 ndmfv 5718 . . . . . . 7  |-  ( -.  A  e.  dom  card  -> 
( card `  A )  =  (/) )
119, 10nsyl4 136 . . . . . 6  |-  ( -.  ( card `  A
)  =  (/)  ->  ( card `  A )  ~~  A )
1211ensymd 7121 . . . . 5  |-  ( -.  ( card `  A
)  =  (/)  ->  A  ~~  ( card `  A
) )
13 breq2 4180 . . . . . 6  |-  ( (
card `  A )  =  ( card `  B
)  ->  ( A  ~~  ( card `  A
)  <->  A  ~~  ( card `  B ) ) )
14 entr 7122 . . . . . . 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 2571   (/)c0 3592   class class class wbr 4176   dom cdm 4841   ` cfv 5417    ~~ cen 7069   cardccrd 7782
This theorem is referenced by:  card1  7815
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 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-er 6868  df-en 7073  df-card 7786
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