Theorem porpss 6941

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.

Step	Hyp	Ref	Expression
1		pssirr 3707	. . . . 5 |- -. x C. x
2		psstr 3711	. . . . 5 |- ( ( x C. y /\ y C. z ) -> x C. z )
3		vex 3203	. . . . . . . 8 |- x e. _V
4	3	brrpss 6940	. . . . . . 7 |- ( x [ C.] x <-> x C. x )
5	4	notbii 310	. . . . . 6 |- ( -. x [ C.] x <-> -. x C. x )
6		vex 3203	. . . . . . . . 9 |- y e. _V
7	6	brrpss 6940	. . . . . . . 8 |- ( x [ C.] y <-> x C. y )
8		vex 3203	. . . . . . . . 9 |- z e. _V
9	8	brrpss 6940	. . . . . . . 8 |- ( y [ C.] z <-> y C. z )
10	7, 9	anbi12i 733	. . . . . . 7 |- ( ( x [ C.] y /\ y [ C.] z ) <-> ( x C. y /\ y C. z ) )
11	8	brrpss 6940	. . . . . . 7 |- ( x [ C.] z <-> x C. z )
12	10, 11	imbi12i 340	. . . . . 6 |- ( ( ( x [ C.] y /\ y [ C.] z ) -> x [ C.] z ) <-> ( ( x C. y /\ y C. z ) -> x C. z ) )
13	5, 12	anbi12i 733	. . . . 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 ) ) )
14	1, 2, 13	mpbir2an 955	. . . 4 |- ( -. x [ C.] x /\ ( ( x [ C.] y /\ y [ C.] z ) -> x [ C.] z ) )
15	14	rgenw 2924	. . 3 |- A. z e. A ( -. x [ C.] x /\ ( ( x [ C.] y /\ y [ C.] z ) -> x [ C.] z ) )
16	15	rgen2w 2925	. 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 5035	. 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 ) ) )
18	16, 17	mpbir 221	1 |- [ C.] Po A

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wral 2912   C. wpss 3575   class class class wbr 4653   Po wpo 5033   [ C.] crpss 6936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-po 5035  df-xp 5120  df-rel 5121  df-rpss 6937

This theorem is referenced by:  sorpss  6942  fin23lem40  9173  isfin1-3  9208  zorng  9326  fin2so  33396