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Theorem carden2b 7690
Description: If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 7689 are meant to replace carden 8263 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 7688 . . . . 5  |-  ( (
card `  B )  e.  ( card `  A
)  ->  -.  ( card `  B )  ~~  A )
2 ennum 7670 . . . . . . . 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 7676 . . . . . . 7  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
53, 4syl 15 . . . . . 6  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  ( card `  B
)  ~~  B )
6 ensym 6998 . . . . . . 7  |-  ( A 
~~  B  ->  B  ~~  A )
76adantr 451 . . . . . 6  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  B  ~~  A )
8 entr 7001 . . . . . 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 7667 . . . . 5  |-  ( card `  A )  e.  On
12 cardon 7667 . . . . 5  |-  ( card `  B )  e.  On
13 ontri1 4508 . . . . 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 7688 . . . . 5  |-  ( (
card `  A )  e.  ( card `  B
)  ->  -.  ( card `  A )  ~~  B )
17 cardid2 7676 . . . . . 6  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
18 id 19 . . . . . 6  |-  ( A 
~~  B  ->  A  ~~  B )
19 entr 7001 . . . . . 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 4508 . . . . 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 3272 . 2  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  ( card `  A
)  =  ( card `  B ) )
26 ndmfv 5635 . . . 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 5635 . . . 4  |-  ( -.  B  e.  dom  card  -> 
( card `  B )  =  (/) )
3129, 30syl 15 . . 3  |-  ( ( A  ~~  B  /\  -.  A  e.  dom  card )  ->  ( card `  B )  =  (/) )
3227, 31eqtr4d 2393 . 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 1642    e. wcel 1710    C_ wss 3228   (/)c0 3531   class class class wbr 4104   Oncon0 4474   dom cdm 4771   ` cfv 5337    ~~ cen 6948   cardccrd 7658
This theorem is referenced by:  card1  7691  carddom2  7700  cardennn  7706  cardsucinf  7707  pm54.43lem  7722  nnacda  7917  ficardun  7918  ackbij1lem5  7940  ackbij1lem8  7943  ackbij1lem9  7944  ackbij2lem2  7956  carden  8263  r1tskina  8494  cardfz  11124
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-er 6747  df-en 6952  df-card 7662
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