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Theorem porpss 6528
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 3449 . . . . 5  |-  -.  x  C.  x
2 psstr 3453 . . . . 5  |-  ( ( x  C.  y  /\  y  C.  z )  ->  x  C.  z )
3 vex 2961 . . . . . . . 8  |-  x  e. 
_V
43brrpss 6527 . . . . . . 7  |-  ( x [
C.]  x  <->  x  C.  x )
54notbii 289 . . . . . 6  |-  ( -.  x [ C.]  x  <->  -.  x  C.  x )
6 vex 2961 . . . . . . . . 9  |-  y  e. 
_V
76brrpss 6527 . . . . . . . 8  |-  ( x [
C.]  y  <->  x  C.  y )
8 vex 2961 . . . . . . . . 9  |-  z  e. 
_V
98brrpss 6527 . . . . . . . 8  |-  ( y [
C.]  z  <->  y  C.  z )
107, 9anbi12i 680 . . . . . . 7  |-  ( ( x [ C.]  y  /\  y [ C.]  z )  <->  ( x  C.  y  /\  y  C.  z ) )
118brrpss 6527 . . . . . . 7  |-  ( x [
C.]  z  <->  x  C.  z )
1210, 11imbi12i 318 . . . . . 6  |-  ( ( ( x [ C.]  y  /\  y [ C.]  z )  ->  x [ C.]  z
)  <->  ( ( x 
C.  y  /\  y  C.  z )  ->  x  C.  z ) )
135, 12anbi12i 680 . . . . 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 888 . . . 4  |-  ( -.  x [ C.]  x  /\  ( ( x [ C.]  y  /\  y [ C.]  z
)  ->  x [ C.]  z ) )
1514rgenw 2775 . . 3  |-  A. z  e.  A  ( -.  x [ C.]  x  /\  (
( x [ C.]  y  /\  y [ C.]  z )  ->  x [ C.]  z
) )
1615rgen2w 2776 . 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 4505 . 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 202 1  |- [ C.]  Po  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360   A.wral 2707    C. wpss 3323   class class class wbr 4214    Po wpo 4503   [ C.] crpss 6523
This theorem is referenced by:  sorpss  6529  fin23lem40  8233  isfin1-3  8268  zorng  8386
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-po 4505  df-xp 4886  df-rel 4887  df-rpss 6524
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