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Theorem carden2b 7810
Description: If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 7809 are meant to replace carden 8382 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 7808 . . . . 5  |-  ( (
card `  B )  e.  ( card `  A
)  ->  -.  ( card `  B )  ~~  A )
2 ennum 7790 . . . . . . . 8  |-  ( A 
~~  B  ->  ( A  e.  dom  card  <->  B  e.  dom  card ) )
32biimpa 471 . . . . . . 7  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  B  e.  dom  card )
4 cardid2 7796 . . . . . . 7  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
53, 4syl 16 . . . . . 6  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  ( card `  B
)  ~~  B )
6 ensym 7115 . . . . . . 7  |-  ( A 
~~  B  ->  B  ~~  A )
76adantr 452 . . . . . 6  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  B  ~~  A )
8 entr 7118 . . . . . 6  |-  ( ( ( card `  B
)  ~~  B  /\  B  ~~  A )  -> 
( card `  B )  ~~  A )
95, 7, 8syl2anc 643 . . . . 5  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  ( card `  B
)  ~~  A )
101, 9nsyl3 113 . . . 4  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  -.  ( card `  B
)  e.  ( card `  A ) )
11 cardon 7787 . . . . 5  |-  ( card `  A )  e.  On
12 cardon 7787 . . . . 5  |-  ( card `  B )  e.  On
13 ontri1 4575 . . . . 5  |-  ( ( ( card `  A
)  e.  On  /\  ( card `  B )  e.  On )  ->  (
( card `  A )  C_  ( card `  B
)  <->  -.  ( card `  B )  e.  (
card `  A )
) )
1411, 12, 13mp2an 654 . . . 4  |-  ( (
card `  A )  C_  ( card `  B
)  <->  -.  ( card `  B )  e.  (
card `  A )
)
1510, 14sylibr 204 . . 3  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  ( card `  A
)  C_  ( card `  B ) )
16 cardne 7808 . . . . 5  |-  ( (
card `  A )  e.  ( card `  B
)  ->  -.  ( card `  A )  ~~  B )
17 cardid2 7796 . . . . . 6  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
18 id 20 . . . . . 6  |-  ( A 
~~  B  ->  A  ~~  B )
19 entr 7118 . . . . . 6  |-  ( ( ( card `  A
)  ~~  A  /\  A  ~~  B )  -> 
( card `  A )  ~~  B )
2017, 18, 19syl2anr 465 . . . . 5  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  ( card `  A
)  ~~  B )
2116, 20nsyl3 113 . . . 4  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  -.  ( card `  A
)  e.  ( card `  B ) )
22 ontri1 4575 . . . . 5  |-  ( ( ( card `  B
)  e.  On  /\  ( card `  A )  e.  On )  ->  (
( card `  B )  C_  ( card `  A
)  <->  -.  ( card `  A )  e.  (
card `  B )
) )
2312, 11, 22mp2an 654 . . . 4  |-  ( (
card `  B )  C_  ( card `  A
)  <->  -.  ( card `  A )  e.  (
card `  B )
)
2421, 23sylibr 204 . . 3  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  ( card `  B
)  C_  ( card `  A ) )
2515, 24eqssd 3325 . 2  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  ( card `  A
)  =  ( card `  B ) )
26 ndmfv 5714 . . . 4  |-  ( -.  A  e.  dom  card  -> 
( card `  A )  =  (/) )
2726adantl 453 . . 3  |-  ( ( A  ~~  B  /\  -.  A  e.  dom  card )  ->  ( card `  A )  =  (/) )
282notbid 286 . . . . 5  |-  ( A 
~~  B  ->  ( -.  A  e.  dom  card  <->  -.  B  e.  dom  card ) )
2928biimpa 471 . . . 4  |-  ( ( A  ~~  B  /\  -.  A  e.  dom  card )  ->  -.  B  e.  dom  card )
30 ndmfv 5714 . . . 4  |-  ( -.  B  e.  dom  card  -> 
( card `  B )  =  (/) )
3129, 30syl 16 . . 3  |-  ( ( A  ~~  B  /\  -.  A  e.  dom  card )  ->  ( card `  B )  =  (/) )
3227, 31eqtr4d 2439 . 2  |-  ( ( A  ~~  B  /\  -.  A  e.  dom  card )  ->  ( card `  A )  =  (
card `  B )
)
3325, 32pm2.61dan 767 1  |-  ( A 
~~  B  ->  ( card `  A )  =  ( card `  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    C_ wss 3280   (/)c0 3588   class class class wbr 4172   Oncon0 4541   dom cdm 4837   ` cfv 5413    ~~ cen 7065   cardccrd 7778
This theorem is referenced by:  card1  7811  carddom2  7820  cardennn  7826  cardsucinf  7827  pm54.43lem  7842  nnacda  8037  ficardun  8038  ackbij1lem5  8060  ackbij1lem8  8063  ackbij1lem9  8064  ackbij2lem2  8076  carden  8382  r1tskina  8613  cardfz  11264
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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