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Theorem porpss 6579
Description: Every class is partially ordered by proper subsets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Assertion
Ref Expression
porpss  |- [ C.]  Po  A

Proof of Theorem porpss
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pssirr 3609 . . . . 5  |-  -.  x  C.  x
2 psstr 3613 . . . . 5  |-  ( ( x  C.  y  /\  y  C.  z )  ->  x  C.  z )
3 vex 3121 . . . . . . . 8  |-  x  e. 
_V
43brrpss 6578 . . . . . . 7  |-  ( x [ C.]  x  <->  x  C.  x )
54notbii 296 . . . . . 6  |-  ( -.  x [ C.]  x  <->  -.  x  C.  x )
6 vex 3121 . . . . . . . . 9  |-  y  e. 
_V
76brrpss 6578 . . . . . . . 8  |-  ( x [ C.]  y  <->  x  C.  y )
8 vex 3121 . . . . . . . . 9  |-  z  e. 
_V
98brrpss 6578 . . . . . . . 8  |-  ( y [ C.]  z  <->  y  C.  z
)
107, 9anbi12i 697 . . . . . . 7  |-  ( ( x [ C.]  y  /\  y [ C.]  z )  <->  ( x  C.  y  /\  y  C.  z ) )
118brrpss 6578 . . . . . . 7  |-  ( x [ C.]  z  <->  x  C.  z )
1210, 11imbi12i 326 . . . . . 6  |-  ( ( ( x [ C.]  y  /\  y [ C.]  z )  ->  x [ C.]  z
)  <->  ( ( x 
C.  y  /\  y  C.  z )  ->  x  C.  z ) )
135, 12anbi12i 697 . . . . 5  |-  ( ( -.  x [ C.]  x  /\  ( ( x [ C.]  y  /\  y [ C.]  z
)  ->  x [ C.]  z
) )  <->  ( -.  x  C.  x  /\  (
( x  C.  y  /\  y  C.  z )  ->  x  C.  z
) ) )
141, 2, 13mpbir2an 918 . . . 4  |-  ( -.  x [ C.]  x  /\  ( ( x [ C.]  y  /\  y [ C.]  z
)  ->  x [ C.]  z
) )
1514rgenw 2828 . . 3  |-  A. z  e.  A  ( -.  x [ C.]  x  /\  (
( x [ C.]  y  /\  y [ C.]  z )  ->  x [ C.]  z
) )
1615rgen2w 2829 . 2  |-  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( -.  x [ C.]  x  /\  (
( x [ C.]  y  /\  y [ C.]  z )  ->  x [ C.]  z
) )
17 df-po 4806 . 2  |-  ( [ C.]  Po  A  <->  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( -.  x [ C.]  x  /\  ( ( x [ C.]  y  /\  y [ C.]  z
)  ->  x [ C.]  z
) ) )
1816, 17mpbir 209 1  |- [ C.]  Po  A
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369   A.wral 2817    C. wpss 3482   class class class wbr 4453    Po wpo 4804   [ C.] crpss 6574
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
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 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-po 4806  df-xp 5011  df-rel 5012  df-rpss 6575
This theorem is referenced by:  sorpss  6580  fin23lem40  8743  isfin1-3  8778  zorng  8896  fin2so  29967
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