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Theorem porpss 6323
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 3310 . . . . 5  |-  -.  x  C.  x
2 psstr 3314 . . . . 5  |-  ( ( x  C.  y  /\  y  C.  z )  ->  x  C.  z )
3 vex 2825 . . . . . . . 8  |-  x  e. 
_V
43brrpss 6322 . . . . . . 7  |-  ( x [
C.]  x  <->  x  C.  x )
54notbii 287 . . . . . 6  |-  ( -.  x [ C.]  x  <->  -.  x  C.  x )
6 vex 2825 . . . . . . . . 9  |-  y  e. 
_V
76brrpss 6322 . . . . . . . 8  |-  ( x [
C.]  y  <->  x  C.  y )
8 vex 2825 . . . . . . . . 9  |-  z  e. 
_V
98brrpss 6322 . . . . . . . 8  |-  ( y [
C.]  z  <->  y  C.  z )
107, 9anbi12i 678 . . . . . . 7  |-  ( ( x [ C.]  y  /\  y [ C.]  z )  <->  ( x  C.  y  /\  y  C.  z ) )
118brrpss 6322 . . . . . . 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 2644 . . 3  |-  A. z  e.  A  ( -.  x [ C.]  x  /\  (
( x [ C.]  y  /\  y [ C.]  z )  ->  x [ C.]  z
) )
1615rgen2w 2645 . 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 4351 . 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 2577    C. wpss 3187   class class class wbr 4060    Po wpo 4349   [ C.] crpss 6318
This theorem is referenced by:  sorpss  6324  fin23lem40  8022  isfin1-3  8057  zorng  8176
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-br 4061  df-opab 4115  df-po 4351  df-xp 4732  df-rel 4733  df-rpss 6319
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