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Theorem sbthcl 7639
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 7521 . 2  |-  Rel  ~~
2 inss1 3718 . . 3  |-  (  ~<_  i^i  `' 
~<_  )  C_  ~<_
3 reldom 7522 . . 3  |-  Rel  ~<_
4 relss 5090 . . 3  |-  ( (  ~<_  i^i  `'  ~<_  )  C_  ~<_  ->  ( Rel  ~<_  ->  Rel  (  ~<_  i^i  `'  ~<_  ) ) )
52, 3, 4mp2 9 . 2  |-  Rel  (  ~<_  i^i  `' 
~<_  )
6 brin 4496 . . 3  |-  ( x (  ~<_  i^i  `'  ~<_  ) y  <-> 
( x  ~<_  y  /\  x `'  ~<_  y )
)
7 vex 3116 . . . . 5  |-  x  e. 
_V
8 vex 3116 . . . . 5  |-  y  e. 
_V
97, 8brcnv 5185 . . . 4  |-  ( x `' 
~<_  y  <->  y  ~<_  x )
109anbi2i 694 . . 3  |-  ( ( x  ~<_  y  /\  x `' 
~<_  y )  <->  ( x  ~<_  y  /\  y  ~<_  x ) )
11 sbthb 7638 . . 3  |-  ( ( x  ~<_  y  /\  y  ~<_  x )  <->  x  ~~  y )
126, 10, 113bitrri 272 . 2  |-  ( x 
~~  y  <->  x (  ~<_  i^i  `' 
~<_  ) y )
131, 5, 12eqbrriv 5098 1  |-  ~~  =  (  ~<_  i^i  `'  ~<_  )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379    i^i cin 3475    C_ wss 3476   class class class wbr 4447   `'ccnv 4998   Rel wrel 5004    ~~ cen 7513    ~<_ cdom 7514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-er 7311  df-en 7517  df-dom 7518
This theorem is referenced by:  dfsdom2  7640
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