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Theorem carden2b 7600
Description: If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 7599 are meant to replace carden 8173 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
Assertion
Ref Expression
carden2b  |-  ( A 
~~  B  ->  ( card `  A )  =  ( card `  B
) )

Proof of Theorem carden2b
StepHypRef Expression
1 cardne 7598 . . . . 5  |-  ( (
card `  B )  e.  ( card `  A
)  ->  -.  ( card `  B )  ~~  A )
2 ennum 7580 . . . . . . . 8  |-  ( A 
~~  B  ->  ( A  e.  dom  card  <->  B  e.  dom  card ) )
32biimpa 470 . . . . . . 7  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  B  e.  dom  card )
4 cardid2 7586 . . . . . . 7  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
53, 4syl 15 . . . . . 6  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  ( card `  B
)  ~~  B )
6 ensym 6910 . . . . . . 7  |-  ( A 
~~  B  ->  B  ~~  A )
76adantr 451 . . . . . 6  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  B  ~~  A )
8 entr 6913 . . . . . 6  |-  ( ( ( card `  B
)  ~~  B  /\  B  ~~  A )  -> 
( card `  B )  ~~  A )
95, 7, 8syl2anc 642 . . . . 5  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  ( card `  B
)  ~~  A )
101, 9nsyl3 111 . . . 4  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  -.  ( card `  B
)  e.  ( card `  A ) )
11 cardon 7577 . . . . 5  |-  ( card `  A )  e.  On
12 cardon 7577 . . . . 5  |-  ( card `  B )  e.  On
13 ontri1 4426 . . . . 5  |-  ( ( ( card `  A
)  e.  On  /\  ( card `  B )  e.  On )  ->  (
( card `  A )  C_  ( card `  B
)  <->  -.  ( card `  B )  e.  (
card `  A )
) )
1411, 12, 13mp2an 653 . . . 4  |-  ( (
card `  A )  C_  ( card `  B
)  <->  -.  ( card `  B )  e.  (
card `  A )
)
1510, 14sylibr 203 . . 3  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  ( card `  A
)  C_  ( card `  B ) )
16 cardne 7598 . . . . 5  |-  ( (
card `  A )  e.  ( card `  B
)  ->  -.  ( card `  A )  ~~  B )
17 cardid2 7586 . . . . . 6  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
18 id 19 . . . . . 6  |-  ( A 
~~  B  ->  A  ~~  B )
19 entr 6913 . . . . . 6  |-  ( ( ( card `  A
)  ~~  A  /\  A  ~~  B )  -> 
( card `  A )  ~~  B )
2017, 18, 19syl2anr 464 . . . . 5  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  ( card `  A
)  ~~  B )
2116, 20nsyl3 111 . . . 4  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  -.  ( card `  A
)  e.  ( card `  B ) )
22 ontri1 4426 . . . . 5  |-  ( ( ( card `  B
)  e.  On  /\  ( card `  A )  e.  On )  ->  (
( card `  B )  C_  ( card `  A
)  <->  -.  ( card `  A )  e.  (
card `  B )
) )
2312, 11, 22mp2an 653 . . . 4  |-  ( (
card `  B )  C_  ( card `  A
)  <->  -.  ( card `  A )  e.  (
card `  B )
)
2421, 23sylibr 203 . . 3  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  ( card `  B
)  C_  ( card `  A ) )
2515, 24eqssd 3196 . 2  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  ( card `  A
)  =  ( card `  B ) )
26 ndmfv 5552 . . . 4  |-  ( -.  A  e.  dom  card  -> 
( card `  A )  =  (/) )
2726adantl 452 . . 3  |-  ( ( A  ~~  B  /\  -.  A  e.  dom  card )  ->  ( card `  A )  =  (/) )
282notbid 285 . . . . 5  |-  ( A 
~~  B  ->  ( -.  A  e.  dom  card  <->  -.  B  e.  dom  card ) )
2928biimpa 470 . . . 4  |-  ( ( A  ~~  B  /\  -.  A  e.  dom  card )  ->  -.  B  e.  dom  card )
30 ndmfv 5552 . . . 4  |-  ( -.  B  e.  dom  card  -> 
( card `  B )  =  (/) )
3129, 30syl 15 . . 3  |-  ( ( A  ~~  B  /\  -.  A  e.  dom  card )  ->  ( card `  B )  =  (/) )
3227, 31eqtr4d 2318 . 2  |-  ( ( A  ~~  B  /\  -.  A  e.  dom  card )  ->  ( card `  A )  =  (
card `  B )
)
3325, 32pm2.61dan 766 1  |-  ( A 
~~  B  ->  ( card `  A )  =  ( card `  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   (/)c0 3455   class class class wbr 4023   Oncon0 4392   dom cdm 4689   ` cfv 5255    ~~ cen 6860   cardccrd 7568
This theorem is referenced by:  card1  7601  carddom2  7610  cardennn  7616  cardsucinf  7617  pm54.43lem  7632  nnacda  7827  ficardun  7828  ackbij1lem5  7850  ackbij1lem8  7853  ackbij1lem9  7854  ackbij2lem2  7866  carden  8173  r1tskina  8404  cardfz  11032
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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