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Theorem sbthcl 7703
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 7585 . 2  |-  Rel  ~~
2 inss1 3682 . . 3  |-  (  ~<_  i^i  `' 
~<_  )  C_  ~<_
3 reldom 7586 . . 3  |-  Rel  ~<_
4 relss 4941 . . 3  |-  ( (  ~<_  i^i  `'  ~<_  )  C_  ~<_  ->  ( Rel  ~<_  ->  Rel  (  ~<_  i^i  `'  ~<_  ) ) )
52, 3, 4mp2 9 . 2  |-  Rel  (  ~<_  i^i  `' 
~<_  )
6 brin 4473 . . 3  |-  ( x (  ~<_  i^i  `'  ~<_  ) y  <-> 
( x  ~<_  y  /\  x `'  ~<_  y )
)
7 vex 3083 . . . . 5  |-  x  e. 
_V
8 vex 3083 . . . . 5  |-  y  e. 
_V
97, 8brcnv 5036 . . . 4  |-  ( x `' 
~<_  y  <->  y  ~<_  x )
109anbi2i 698 . . 3  |-  ( ( x  ~<_  y  /\  x `' 
~<_  y )  <->  ( x  ~<_  y  /\  y  ~<_  x ) )
11 sbthb 7702 . . 3  |-  ( ( x  ~<_  y  /\  y  ~<_  x )  <->  x  ~~  y )
126, 10, 113bitrri 275 . 2  |-  ( x 
~~  y  <->  x (  ~<_  i^i  `' 
~<_  ) y )
131, 5, 12eqbrriv 4949 1  |-  ~~  =  (  ~<_  i^i  `'  ~<_  )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    = wceq 1437    i^i cin 3435    C_ wss 3436   class class class wbr 4423   `'ccnv 4852   Rel wrel 4858    ~~ cen 7577    ~<_ cdom 7578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-er 7374  df-en 7581  df-dom 7582
This theorem is referenced by:  dfsdom2  7704
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