MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  en2i Unicode version

Theorem en2i 7108
Description: Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 4-Jan-2004.)
Hypotheses
Ref Expression
en2i.1  |-  A  e. 
_V
en2i.2  |-  B  e. 
_V
en2i.3  |-  ( x  e.  A  ->  C  e.  _V )
en2i.4  |-  ( y  e.  B  ->  D  e.  _V )
en2i.5  |-  ( ( x  e.  A  /\  y  =  C )  <->  ( y  e.  B  /\  x  =  D )
)
Assertion
Ref Expression
en2i  |-  A  ~~  B
Distinct variable groups:    x, y, A    x, B, y    y, C    x, D
Allowed substitution hints:    C( x)    D( y)

Proof of Theorem en2i
StepHypRef Expression
1 en2i.1 . . . 4  |-  A  e. 
_V
21a1i 11 . . 3  |-  (  T. 
->  A  e.  _V )
3 en2i.2 . . . 4  |-  B  e. 
_V
43a1i 11 . . 3  |-  (  T. 
->  B  e.  _V )
5 en2i.3 . . . 4  |-  ( x  e.  A  ->  C  e.  _V )
65a1i 11 . . 3  |-  (  T. 
->  ( x  e.  A  ->  C  e.  _V )
)
7 en2i.4 . . . 4  |-  ( y  e.  B  ->  D  e.  _V )
87a1i 11 . . 3  |-  (  T. 
->  ( y  e.  B  ->  D  e.  _V )
)
9 en2i.5 . . . 4  |-  ( ( x  e.  A  /\  y  =  C )  <->  ( y  e.  B  /\  x  =  D )
)
109a1i 11 . . 3  |-  (  T. 
->  ( ( x  e.  A  /\  y  =  C )  <->  ( y  e.  B  /\  x  =  D ) ) )
112, 4, 6, 8, 10en2d 7106 . 2  |-  (  T. 
->  A  ~~  B )
1211trud 1329 1  |-  A  ~~  B
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    T. wtru 1322    = wceq 1649    e. wcel 1721   _Vcvv 2920   class class class wbr 4176    ~~ cen 7069
This theorem is referenced by:  mapsnen  7147  xpsnen  7155  xpassen  7165
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-en 7073
  Copyright terms: Public domain W3C validator