MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  enen2 Structured version   Unicode version

Theorem enen2 7450
Description: Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
Assertion
Ref Expression
enen2  |-  ( A 
~~  B  ->  ( C  ~~  A  <->  C  ~~  B ) )

Proof of Theorem enen2
StepHypRef Expression
1 entr 7359 . . 3  |-  ( ( C  ~~  A  /\  A  ~~  B )  ->  C  ~~  B )
21ancoms 453 . 2  |-  ( ( A  ~~  B  /\  C  ~~  A )  ->  C  ~~  B )
3 ensym 7356 . . 3  |-  ( A 
~~  B  ->  B  ~~  A )
4 entr 7359 . . . 4  |-  ( ( C  ~~  B  /\  B  ~~  A )  ->  C  ~~  A )
54ancoms 453 . . 3  |-  ( ( B  ~~  A  /\  C  ~~  B )  ->  C  ~~  A )
63, 5sylan 471 . 2  |-  ( ( A  ~~  B  /\  C  ~~  B )  ->  C  ~~  A )
72, 6impbida 828 1  |-  ( A 
~~  B  ->  ( C  ~~  A  <->  C  ~~  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   class class class wbr 4290    ~~ cen 7305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-er 7099  df-en 7309
This theorem is referenced by:  karden  8100  ennum  8115  pwcdaen  8352  alephexp1  8741  gchdomtri  8794  gch-kn  8842  ctbnfien  29154
  Copyright terms: Public domain W3C validator