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Theorem porpss 6385
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 3477 . . . . 5  |-  -.  x  C.  x
2 psstr 3481 . . . . 5  |-  ( ( x  C.  y  /\  y  C.  z )  ->  x  C.  z )
3 vex 2996 . . . . . . . 8  |-  x  e. 
_V
43brrpss 6384 . . . . . . 7  |-  ( x [
C.]  x  <->  x  C.  x
)
54notbii 296 . . . . . 6  |-  ( -.  x [ C.]  x  <->  -.  x  C.  x )
6 vex 2996 . . . . . . . . 9  |-  y  e. 
_V
76brrpss 6384 . . . . . . . 8  |-  ( x [
C.]  y  <->  x  C.  y
)
8 vex 2996 . . . . . . . . 9  |-  z  e. 
_V
98brrpss 6384 . . . . . . . 8  |-  ( y [
C.]  z  <->  y  C.  z
)
107, 9anbi12i 697 . . . . . . 7  |-  ( ( x [ C.]  y  /\  y [ C.]  z )  <->  ( x  C.  y  /\  y  C.  z ) )
118brrpss 6384 . . . . . . 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 911 . . . 4  |-  ( -.  x [ C.]  x  /\  ( ( x [ C.]  y  /\  y [ C.]  z
)  ->  x [ C.]  z ) )
1514rgenw 2804 . . 3  |-  A. z  e.  A  ( -.  x [ C.]  x  /\  (
( x [ C.]  y  /\  y [ C.]  z )  ->  x [ C.]  z
) )
1615rgen2w 2805 . 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 4662 . 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 2736    C. wpss 3350   class class class wbr 4313    Po wpo 4660   [ C.] crpss 6380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pr 4552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-br 4314  df-opab 4372  df-po 4662  df-xp 4867  df-rel 4868  df-rpss 6381
This theorem is referenced by:  sorpss  6386  fin23lem40  8541  isfin1-3  8576  zorng  8694  fin2so  28442
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