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Theorem sbthcl 7232
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 7117 . 2  |-  Rel  ~~
2 inss1 3563 . . 3  |-  (  ~<_  i^i  `' 
~<_  )  C_  ~<_
3 reldom 7118 . . 3  |-  Rel  ~<_
4 relss 4966 . . 3  |-  ( (  ~<_  i^i  `'  ~<_  )  C_  ~<_  ->  ( Rel  ~<_  ->  Rel  (  ~<_  i^i  `'  ~<_  ) ) )
52, 3, 4mp2 9 . 2  |-  Rel  (  ~<_  i^i  `' 
~<_  )
6 brin 4262 . . 3  |-  ( x (  ~<_  i^i  `'  ~<_  ) y  <-> 
( x  ~<_  y  /\  x `'  ~<_  y )
)
7 vex 2961 . . . . 5  |-  x  e. 
_V
8 vex 2961 . . . . 5  |-  y  e. 
_V
97, 8brcnv 5058 . . . 4  |-  ( x `' 
~<_  y  <->  y  ~<_  x )
109anbi2i 677 . . 3  |-  ( ( x  ~<_  y  /\  x `' 
~<_  y )  <->  ( x  ~<_  y  /\  y  ~<_  x ) )
11 sbthb 7231 . . 3  |-  ( ( x  ~<_  y  /\  y  ~<_  x )  <->  x  ~~  y )
126, 10, 113bitrri 265 . 2  |-  ( x 
~~  y  <->  x (  ~<_  i^i  `' 
~<_  ) y )
131, 5, 12eqbrriv 4974 1  |-  ~~  =  (  ~<_  i^i  `'  ~<_  )
Colors of variables: wff set class
Syntax hints:    /\ wa 360    = wceq 1653    i^i cin 3321    C_ wss 3322   class class class wbr 4215   `'ccnv 4880   Rel wrel 4886    ~~ cen 7109    ~<_ cdom 7110
This theorem is referenced by:  dfsdom2  7233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-er 6908  df-en 7113  df-dom 7114
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