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Theorem brrpssg 6295
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 2809 . . 3  |-  ( B  e.  V  ->  B  e.  _V )
2 relrpss 6294 . . . 4  |-  Rel [ C.]
32brrelexi 4745 . . 3  |-  ( A [
C.]  B  ->  A  e.  _V )
41, 3anim12i 549 . 2  |-  ( ( B  e.  V  /\  A [ C.]  B )  -> 
( B  e.  _V  /\  A  e.  _V )
)
51adantr 451 . . 3  |-  ( ( B  e.  V  /\  A  C.  B )  ->  B  e.  _V )
6 pssss 3284 . . . 4  |-  ( A 
C.  B  ->  A  C_  B )
7 ssexg 4176 . . . 4  |-  ( ( A  C_  B  /\  B  e.  _V )  ->  A  e.  _V )
86, 1, 7syl2anr 464 . . 3  |-  ( ( B  e.  V  /\  A  C.  B )  ->  A  e.  _V )
95, 8jca 518 . 2  |-  ( ( B  e.  V  /\  A  C.  B )  -> 
( B  e.  _V  /\  A  e.  _V )
)
10 psseq1 3276 . . . 4  |-  ( x  =  A  ->  (
x  C.  y  <->  A  C.  y ) )
11 psseq2 3277 . . . 4  |-  ( y  =  B  ->  ( A  C.  y  <->  A  C.  B ) )
12 df-rpss 6293 . . . 4  |- [ C.]  =  { <. x ,  y
>.  |  x  C.  y }
1310, 11, 12brabg 4300 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A [ C.]  B  <->  A 
C.  B ) )
1413ancoms 439 . 2  |-  ( ( B  e.  _V  /\  A  e.  _V )  ->  ( A [ C.]  B  <->  A 
C.  B ) )
154, 9, 14pm5.21nd 868 1  |-  ( B  e.  V  ->  ( A [ C.]  B  <->  A  C.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1696   _Vcvv 2801    C_ wss 3165    C. wpss 3166   class class class wbr 4039   [ C.] crpss 6292
This theorem is referenced by:  brrpss  6296  sorpssi  6299
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-rpss 6293
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