MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  carden2a Structured version   Unicode version

Theorem carden2a 7858
Description: If two sets have equal nonzero cardinalities, then they are equinumerous. (This assertion and carden2b 7859 are meant to replace carden 8431 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 2603 . 2  |-  ( (
card `  A )  =/=  (/)  <->  -.  ( card `  A )  =  (/) )
2 ndmfv 5758 . . . . . . 7  |-  ( -.  B  e.  dom  card  -> 
( card `  B )  =  (/) )
3 eqeq1 2444 . . . . . . 7  |-  ( (
card `  A )  =  ( card `  B
)  ->  ( ( card `  A )  =  (/) 
<->  ( card `  B
)  =  (/) ) )
42, 3syl5ibr 214 . . . . . 6  |-  ( (
card `  A )  =  ( card `  B
)  ->  ( -.  B  e.  dom  card  ->  (
card `  A )  =  (/) ) )
54con1d 119 . . . . 5  |-  ( (
card `  A )  =  ( card `  B
)  ->  ( -.  ( card `  A )  =  (/)  ->  B  e.  dom  card ) )
65imp 420 . . . 4  |-  ( ( ( card `  A
)  =  ( card `  B )  /\  -.  ( card `  A )  =  (/) )  ->  B  e.  dom  card )
7 cardid2 7845 . . . 4  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
86, 7syl 16 . . 3  |-  ( ( ( card `  A
)  =  ( card `  B )  /\  -.  ( card `  A )  =  (/) )  ->  ( card `  B )  ~~  B )
9 cardid2 7845 . . . . . . 7  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
10 ndmfv 5758 . . . . . . 7  |-  ( -.  A  e.  dom  card  -> 
( card `  A )  =  (/) )
119, 10nsyl4 137 . . . . . 6  |-  ( -.  ( card `  A
)  =  (/)  ->  ( card `  A )  ~~  A )
1211ensymd 7161 . . . . 5  |-  ( -.  ( card `  A
)  =  (/)  ->  A  ~~  ( card `  A
) )
13 breq2 4219 . . . . . 6  |-  ( (
card `  A )  =  ( card `  B
)  ->  ( A  ~~  ( card `  A
)  <->  A  ~~  ( card `  B ) ) )
14 entr 7162 . . . . . . 7  |-  ( ( A  ~~  ( card `  B )  /\  ( card `  B )  ~~  B )  ->  A  ~~  B )
1514ex 425 . . . . . 6  |-  ( A 
~~  ( card `  B
)  ->  ( ( card `  B )  ~~  B  ->  A  ~~  B
) )
1613, 15syl6bi 221 . . . . 5  |-  ( (
card `  A )  =  ( card `  B
)  ->  ( A  ~~  ( card `  A
)  ->  ( ( card `  B )  ~~  B  ->  A  ~~  B
) ) )
1712, 16syl5 31 . . . 4  |-  ( (
card `  A )  =  ( card `  B
)  ->  ( -.  ( card `  A )  =  (/)  ->  ( ( card `  B )  ~~  B  ->  A  ~~  B
) ) )
1817imp 420 . . 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 463 1  |-  ( ( ( card `  A
)  =  ( card `  B )  /\  ( card `  A )  =/=  (/) )  ->  A  ~~  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   (/)c0 3630   class class class wbr 4215   dom cdm 4881   ` cfv 5457    ~~ cen 7109   cardccrd 7827
This theorem is referenced by:  card1  7860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-er 6908  df-en 7113  df-card 7831
  Copyright terms: Public domain W3C validator