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Theorem sbthcl 7196
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 7081 . 2  |-  Rel  ~~
2 inss1 3529 . . 3  |-  (  ~<_  i^i  `' 
~<_  )  C_  ~<_
3 reldom 7082 . . 3  |-  Rel  ~<_
4 relss 4930 . . 3  |-  ( (  ~<_  i^i  `'  ~<_  )  C_  ~<_  ->  ( Rel  ~<_  ->  Rel  (  ~<_  i^i  `'  ~<_  ) ) )
52, 3, 4mp2 9 . 2  |-  Rel  (  ~<_  i^i  `' 
~<_  )
6 brin 4227 . . 3  |-  ( x (  ~<_  i^i  `'  ~<_  ) y  <-> 
( x  ~<_  y  /\  x `'  ~<_  y )
)
7 vex 2927 . . . . 5  |-  x  e. 
_V
8 vex 2927 . . . . 5  |-  y  e. 
_V
97, 8brcnv 5022 . . . 4  |-  ( x `' 
~<_  y  <->  y  ~<_  x )
109anbi2i 676 . . 3  |-  ( ( x  ~<_  y  /\  x `' 
~<_  y )  <->  ( x  ~<_  y  /\  y  ~<_  x ) )
11 sbthb 7195 . . 3  |-  ( ( x  ~<_  y  /\  y  ~<_  x )  <->  x  ~~  y )
126, 10, 113bitrri 264 . 2  |-  ( x 
~~  y  <->  x (  ~<_  i^i  `' 
~<_  ) y )
131, 5, 12eqbrriv 4938 1  |-  ~~  =  (  ~<_  i^i  `'  ~<_  )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    i^i cin 3287    C_ wss 3288   class class class wbr 4180   `'ccnv 4844   Rel wrel 4850    ~~ cen 7073    ~<_ cdom 7074
This theorem is referenced by:  dfsdom2  7197
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 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-er 6872  df-en 7077  df-dom 7078
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