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Theorem carden2 8426
Description: Two numerable sets are equinumerous iff their cardinal numbers are equal. Unlike carden 8981, the Axiom of Choice is not required. (Contributed by Mario Carneiro, 22-Sep-2013.)
Assertion
Ref Expression
carden2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  =  (
card `  B )  <->  A 
~~  B ) )

Proof of Theorem carden2
StepHypRef Expression
1 carddom2 8416 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  C_  ( card `  B )  <->  A  ~<_  B ) )
2 carddom2 8416 . . . 4  |-  ( ( B  e.  dom  card  /\  A  e.  dom  card )  ->  ( ( card `  B )  C_  ( card `  A )  <->  B  ~<_  A ) )
32ancoms 455 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  B )  C_  ( card `  A )  <->  B  ~<_  A ) )
41, 3anbi12d 718 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( (
card `  A )  C_  ( card `  B
)  /\  ( card `  B )  C_  ( card `  A ) )  <-> 
( A  ~<_  B  /\  B  ~<_  A ) ) )
5 eqss 3449 . . 3  |-  ( (
card `  A )  =  ( card `  B
)  <->  ( ( card `  A )  C_  ( card `  B )  /\  ( card `  B )  C_  ( card `  A
) ) )
65bicomi 206 . 2  |-  ( ( ( card `  A
)  C_  ( card `  B )  /\  ( card `  B )  C_  ( card `  A )
)  <->  ( card `  A
)  =  ( card `  B ) )
7 sbthb 7698 . 2  |-  ( ( A  ~<_  B  /\  B  ~<_  A )  <->  A  ~~  B )
84, 6, 73bitr3g 291 1  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  =  (
card `  B )  <->  A 
~~  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1446    e. wcel 1889    C_ wss 3406   class class class wbr 4405   dom cdm 4837   ` cfv 5585    ~~ cen 7571    ~<_ cdom 7572   cardccrd 8374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  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-ord 5429  df-on 5430  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-card 8378
This theorem is referenced by:  cardsdom2  8427  pm54.43lem  8438  sdom2en01  8737  fin23lem22  8762  fin1a2lem9  8843  pwfseqlem4  9092  hashen  12537
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