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Theorem porpss 6281
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 3276 . . . . 5  |-  -.  x  C.  x
2 psstr 3280 . . . . 5  |-  ( ( x  C.  y  /\  y  C.  z )  ->  x  C.  z )
3 vex 2791 . . . . . . . 8  |-  x  e. 
_V
43brrpss 6280 . . . . . . 7  |-  ( x [
C.]  x  <->  x  C.  x )
54notbii 287 . . . . . 6  |-  ( -.  x [ C.]  x  <->  -.  x  C.  x )
6 vex 2791 . . . . . . . . 9  |-  y  e. 
_V
76brrpss 6280 . . . . . . . 8  |-  ( x [
C.]  y  <->  x  C.  y )
8 vex 2791 . . . . . . . . 9  |-  z  e. 
_V
98brrpss 6280 . . . . . . . 8  |-  ( y [
C.]  z  <->  y  C.  z )
107, 9anbi12i 678 . . . . . . 7  |-  ( ( x [ C.]  y  /\  y [ C.]  z )  <->  ( x  C.  y  /\  y  C.  z ) )
118brrpss 6280 . . . . . . 7  |-  ( x [
C.]  z  <->  x  C.  z )
1210, 11imbi12i 316 . . . . . 6  |-  ( ( ( x [ C.]  y  /\  y [ C.]  z )  ->  x [ C.]  z
)  <->  ( ( x 
C.  y  /\  y  C.  z )  ->  x  C.  z ) )
135, 12anbi12i 678 . . . . 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 886 . . . 4  |-  ( -.  x [ C.]  x  /\  ( ( x [ C.]  y  /\  y [ C.]  z
)  ->  x [ C.]  z ) )
1514rgenw 2610 . . 3  |-  A. z  e.  A  ( -.  x [ C.]  x  /\  (
( x [ C.]  y  /\  y [ C.]  z )  ->  x [ C.]  z
) )
1615rgen2w 2611 . 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 4314 . 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 200 1  |- [ C.]  Po  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wral 2543    C. wpss 3153   class class class wbr 4023    Po wpo 4312   [ C.] crpss 6276
This theorem is referenced by:  sorpss  6282  fin23lem40  7977  isfin1-3  8012  zorng  8131
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-po 4314  df-xp 4695  df-rel 4696  df-rpss 6277
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