MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  carden2 Structured version   Unicode version

Theorem carden2 8359
Description: Two numerable sets are equinumerous iff their cardinal numbers are equal. Unlike carden 8917, 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 8349 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  C_  ( card `  B )  <->  A  ~<_  B ) )
2 carddom2 8349 . . . 4  |-  ( ( B  e.  dom  card  /\  A  e.  dom  card )  ->  ( ( card `  B )  C_  ( card `  A )  <->  B  ~<_  A ) )
32ancoms 453 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  B )  C_  ( card `  A )  <->  B  ~<_  A ) )
41, 3anbi12d 710 . 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 3514 . . 3  |-  ( (
card `  A )  =  ( card `  B
)  <->  ( ( card `  A )  C_  ( card `  B )  /\  ( card `  B )  C_  ( card `  A
) ) )
65bicomi 202 . 2  |-  ( ( ( card `  A
)  C_  ( card `  B )  /\  ( card `  B )  C_  ( card `  A )
)  <->  ( card `  A
)  =  ( card `  B ) )
7 sbthb 7630 . 2  |-  ( ( A  ~<_  B  /\  B  ~<_  A )  <->  A  ~~  B )
84, 6, 73bitr3g 287 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 184    /\ wa 369    = wceq 1374    e. wcel 1762    C_ wss 3471   class class class wbr 4442   dom cdm 4994   ` cfv 5581    ~~ cen 7505    ~<_ cdom 7506   cardccrd 8307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-card 8311
This theorem is referenced by:  cardsdom2  8360  pm54.43lem  8371  sdom2en01  8673  fin23lem22  8698  fin1a2lem9  8779  pwfseqlem4  9031  hashen  12377
  Copyright terms: Public domain W3C validator