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Theorem brrpssg 6465
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 3080 . . 3  |-  ( B  e.  V  ->  B  e.  _V )
2 relrpss 6464 . . . 4  |-  Rel [ C.]
32brrelexi 4980 . . 3  |-  ( A [
C.]  B  ->  A  e.  _V )
41, 3anim12i 566 . 2  |-  ( ( B  e.  V  /\  A [ C.]  B )  -> 
( B  e.  _V  /\  A  e.  _V )
)
51adantr 465 . . 3  |-  ( ( B  e.  V  /\  A  C.  B )  ->  B  e.  _V )
6 pssss 3552 . . . 4  |-  ( A 
C.  B  ->  A  C_  B )
7 ssexg 4539 . . . 4  |-  ( ( A  C_  B  /\  B  e.  _V )  ->  A  e.  _V )
86, 1, 7syl2anr 478 . . 3  |-  ( ( B  e.  V  /\  A  C.  B )  ->  A  e.  _V )
95, 8jca 532 . 2  |-  ( ( B  e.  V  /\  A  C.  B )  -> 
( B  e.  _V  /\  A  e.  _V )
)
10 psseq1 3544 . . . 4  |-  ( x  =  A  ->  (
x  C.  y  <->  A  C.  y
) )
11 psseq2 3545 . . . 4  |-  ( y  =  B  ->  ( A  C.  y  <->  A  C.  B
) )
12 df-rpss 6463 . . . 4  |- [ C.]  =  { <. x ,  y
>.  |  x  C.  y }
1310, 11, 12brabg 4709 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A [ C.]  B  <->  A 
C.  B ) )
1413ancoms 453 . 2  |-  ( ( B  e.  _V  /\  A  e.  _V )  ->  ( A [ C.]  B  <->  A 
C.  B ) )
154, 9, 14pm5.21nd 893 1  |-  ( B  e.  V  ->  ( A [ C.]  B  <->  A  C.  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1758   _Vcvv 3071    C_ wss 3429    C. wpss 3430   class class class wbr 4393   [ C.] crpss 6462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-br 4394  df-opab 4452  df-xp 4947  df-rel 4948  df-rpss 6463
This theorem is referenced by:  brrpss  6466  sorpssi  6469
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