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

Theorem enen2 6999
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 6910 . . 3  |-  ( ( C  ~~  A  /\  A  ~~  B )  ->  C  ~~  B )
21ancoms 441 . 2  |-  ( ( A  ~~  B  /\  C  ~~  A )  ->  C  ~~  B )
3 ensym 6907 . . 3  |-  ( A 
~~  B  ->  B  ~~  A )
4 entr 6910 . . . 4  |-  ( ( C  ~~  B  /\  B  ~~  A )  ->  C  ~~  A )
54ancoms 441 . . 3  |-  ( ( B  ~~  A  /\  C  ~~  B )  ->  C  ~~  A )
63, 5sylan 459 . 2  |-  ( ( A  ~~  B  /\  C  ~~  B )  ->  C  ~~  A )
72, 6impbida 807 1  |-  ( A 
~~  B  ->  ( C  ~~  A  <->  C  ~~  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178   class class class wbr 4026    ~~ cen 6857
This theorem is referenced by:  karden  7562  ennum  7577  pwcdaen  7808  alephexp1  8198  gchdomtri  8248  gch-kn  8300  ctbnfien  26302
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-sep 4144  ax-nul 4152  ax-pow 4189  ax-pr 4215  ax-un 4513
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-ral 2551  df-rex 2552  df-rab 2555  df-v 2793  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-nul 3459  df-if 3569  df-pw 3630  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3831  df-br 4027  df-opab 4081  df-id 4310  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-er 6657  df-en 6861
  Copyright terms: Public domain W3C validator