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Theorem sbthcl 7679
Description: Schroeder-Bernstein Theorem in class form. (Contributed by NM, 28-Mar-1998.)
Assertion
Ref Expression
sbthcl  |-  ~~  =  (  ~<_  i^i  `'  ~<_  )

Proof of Theorem sbthcl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relen 7561 . 2  |-  Rel  ~~
2 inss1 3661 . . 3  |-  (  ~<_  i^i  `' 
~<_  )  C_  ~<_
3 reldom 7562 . . 3  |-  Rel  ~<_
4 relss 4913 . . 3  |-  ( (  ~<_  i^i  `'  ~<_  )  C_  ~<_  ->  ( Rel  ~<_  ->  Rel  (  ~<_  i^i  `'  ~<_  ) ) )
52, 3, 4mp2 9 . 2  |-  Rel  (  ~<_  i^i  `' 
~<_  )
6 brin 4446 . . 3  |-  ( x (  ~<_  i^i  `'  ~<_  ) y  <-> 
( x  ~<_  y  /\  x `'  ~<_  y )
)
7 vex 3064 . . . . 5  |-  x  e. 
_V
8 vex 3064 . . . . 5  |-  y  e. 
_V
97, 8brcnv 5008 . . . 4  |-  ( x `' 
~<_  y  <->  y  ~<_  x )
109anbi2i 694 . . 3  |-  ( ( x  ~<_  y  /\  x `' 
~<_  y )  <->  ( x  ~<_  y  /\  y  ~<_  x ) )
11 sbthb 7678 . . 3  |-  ( ( x  ~<_  y  /\  y  ~<_  x )  <->  x  ~~  y )
126, 10, 113bitrri 274 . 2  |-  ( x 
~~  y  <->  x (  ~<_  i^i  `' 
~<_  ) y )
131, 5, 12eqbrriv 4921 1  |-  ~~  =  (  ~<_  i^i  `'  ~<_  )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1407    i^i cin 3415    C_ wss 3416   class class class wbr 4397   `'ccnv 4824   Rel wrel 4830    ~~ cen 7553    ~<_ cdom 7554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-er 7350  df-en 7557  df-dom 7558
This theorem is referenced by:  dfsdom2  7680
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