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Theorem brrpssg 6555
Description: The proper subset relation on sets is the same as class proper subsethood. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Assertion
Ref Expression
brrpssg  |-  ( B  e.  V  ->  ( A [ C.]  B  <->  A  C.  B
) )

Proof of Theorem brrpssg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3115 . . 3  |-  ( B  e.  V  ->  B  e.  _V )
2 relrpss 6554 . . . 4  |-  Rel [ C.]
32brrelexi 5029 . . 3  |-  ( A [ C.]  B  ->  A  e.  _V )
41, 3anim12i 564 . 2  |-  ( ( B  e.  V  /\  A [ C.]  B )  -> 
( B  e.  _V  /\  A  e.  _V )
)
51adantr 463 . . 3  |-  ( ( B  e.  V  /\  A  C.  B )  ->  B  e.  _V )
6 pssss 3585 . . . 4  |-  ( A 
C.  B  ->  A  C_  B )
7 ssexg 4583 . . . 4  |-  ( ( A  C_  B  /\  B  e.  _V )  ->  A  e.  _V )
86, 1, 7syl2anr 476 . . 3  |-  ( ( B  e.  V  /\  A  C.  B )  ->  A  e.  _V )
95, 8jca 530 . 2  |-  ( ( B  e.  V  /\  A  C.  B )  -> 
( B  e.  _V  /\  A  e.  _V )
)
10 psseq1 3577 . . . 4  |-  ( x  =  A  ->  (
x  C.  y  <->  A  C.  y
) )
11 psseq2 3578 . . . 4  |-  ( y  =  B  ->  ( A  C.  y  <->  A  C.  B
) )
12 df-rpss 6553 . . . 4  |- [ C.]  =  { <. x ,  y
>.  |  x  C.  y }
1310, 11, 12brabg 4755 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A [ C.]  B  <->  A  C.  B
) )
1413ancoms 451 . 2  |-  ( ( B  e.  _V  /\  A  e.  _V )  ->  ( A [ C.]  B  <->  A  C.  B
) )
154, 9, 14pm5.21nd 898 1  |-  ( B  e.  V  ->  ( A [ C.]  B  <->  A  C.  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    e. wcel 1823   _Vcvv 3106    C_ wss 3461    C. wpss 3462   class class class wbr 4439   [ C.] crpss 6552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-rpss 6553
This theorem is referenced by:  brrpss  6556  sorpssi  6559
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