Theorem sorpss 6942

Description: Express strict ordering under proper subsets, i.e. the notion of a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.)

Assertion

Ref	Expression
sorpss	|- ( [ C.] Or A <-> A. x e. A A. y e. A ( x C_ y \/ y C_ x ) )

Distinct variable group:   x, y, A

Proof of Theorem sorpss

Step	Hyp	Ref	Expression
1		porpss 6941	. . 3 |- [ C.] Po A
2	1	biantrur 527	. 2 |- ( A. x e. A A. y e. A ( x [ C.] y \/ x = y \/ y [ C.] x ) <-> ( [ C.] Po A /\ A. x e. A A. y e. A ( x [ C.] y \/ x = y \/ y [ C.] x ) ) )
3		sspsstri 3709	. . . 4 |- ( ( x C_ y \/ y C_ x ) <-> ( x C. y \/ x = y \/ y C. x ) )
4		vex 3203	. . . . . 6 |- y e. _V
5	4	brrpss 6940	. . . . 5 |- ( x [ C.] y <-> x C. y )
6		biid 251	. . . . 5 |- ( x = y <-> x = y )
7		vex 3203	. . . . . 6 |- x e. _V
8	7	brrpss 6940	. . . . 5 |- ( y [ C.] x <-> y C. x )
9	5, 6, 8	3orbi123i 1252	. . . 4 |- ( ( x [ C.] y \/ x = y \/ y [ C.] x ) <-> ( x C. y \/ x = y \/ y C. x ) )
10	3, 9	bitr4i 267	. . 3 |- ( ( x C_ y \/ y C_ x ) <-> ( x [ C.] y \/ x = y \/ y [ C.] x ) )
11	10	2ralbii 2981	. 2 |- ( A. x e. A A. y e. A ( x C_ y \/ y C_ x ) <-> A. x e. A A. y e. A ( x [ C.] y \/ x = y \/ y [ C.] x ) )
12		df-so 5036	. 2 |- ( [ C.] Or A <-> ( [ C.] Po A /\ A. x e. A A. y e. A ( x [ C.] y \/ x = y \/ y [ C.] x ) ) )
13	2, 11, 12	3bitr4ri 293	1 |- ( [ C.] Or A <-> A. x e. A A. y e. A ( x C_ y \/ y C_ x ) )

Syntax hints:   <-> wb 196   \/ wo 383   /\ wa 384   \/ w3o 1036   A.wral 2912   C_ wss 3574   C. wpss 3575   class class class wbr 4653   Po wpo 5033   Or wor 5034   [ 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-3or 1038  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-so 5036  df-xp 5120  df-rel 5121  df-rpss 6937

This theorem is referenced by:  sorpsscmpl  6948  enfin2i  9143  fin1a2lem13  9234