Theorem psslinpr 9853

Description: Proper subset is a linear ordering on positive reals. Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 25-Feb-1996.) (New usage is discouraged.)

Assertion

Ref	Expression
psslinpr	|- ( ( A e. P. /\ B e. P. ) -> ( A C. B \/ A = B \/ B C. A ) )

Proof of Theorem psslinpr

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elprnq 9813	. . . . . . . . . . . . 13 |- ( ( A e. P. /\ x e. A ) -> x e. Q. )
2		prub 9816	. . . . . . . . . . . . 13 |- ( ( ( B e. P. /\ y e. B ) /\ x e. Q. ) -> ( -. x e. B -> y <Q x ) )
3	1, 2	sylan2 491	. . . . . . . . . . . 12 |- ( ( ( B e. P. /\ y e. B ) /\ ( A e. P. /\ x e. A ) ) -> ( -. x e. B -> y <Q x ) )
4		prcdnq 9815	. . . . . . . . . . . . 13 |- ( ( A e. P. /\ x e. A ) -> ( y <Q x -> y e. A ) )
5	4	adantl 482	. . . . . . . . . . . 12 |- ( ( ( B e. P. /\ y e. B ) /\ ( A e. P. /\ x e. A ) ) -> ( y <Q x -> y e. A ) )
6	3, 5	syld 47	. . . . . . . . . . 11 |- ( ( ( B e. P. /\ y e. B ) /\ ( A e. P. /\ x e. A ) ) -> ( -. x e. B -> y e. A ) )
7	6	exp43 640	. . . . . . . . . 10 |- ( B e. P. -> ( y e. B -> ( A e. P. -> ( x e. A -> ( -. x e. B -> y e. A ) ) ) ) )
8	7	com3r 87	. . . . . . . . 9 |- ( A e. P. -> ( B e. P. -> ( y e. B -> ( x e. A -> ( -. x e. B -> y e. A ) ) ) ) )
9	8	imp 445	. . . . . . . 8 |- ( ( A e. P. /\ B e. P. ) -> ( y e. B -> ( x e. A -> ( -. x e. B -> y e. A ) ) ) )
10	9	imp4a 614	. . . . . . 7 |- ( ( A e. P. /\ B e. P. ) -> ( y e. B -> ( ( x e. A /\ -. x e. B ) -> y e. A ) ) )
11	10	com23 86	. . . . . 6 |- ( ( A e. P. /\ B e. P. ) -> ( ( x e. A /\ -. x e. B ) -> ( y e. B -> y e. A ) ) )
12	11	alrimdv 1857	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( ( x e. A /\ -. x e. B ) -> A. y ( y e. B -> y e. A ) ) )
13	12	exlimdv 1861	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( E. x ( x e. A /\ -. x e. B ) -> A. y ( y e. B -> y e. A ) ) )
14		nss 3663	. . . . 5 |- ( -. A C_ B <-> E. x ( x e. A /\ -. x e. B ) )
15		sspss 3706	. . . . 5 |- ( A C_ B <-> ( A C. B \/ A = B ) )
16	14, 15	xchnxbi 322	. . . 4 |- ( -. ( A C. B \/ A = B ) <-> E. x ( x e. A /\ -. x e. B ) )
17		sspss 3706	. . . . 5 |- ( B C_ A <-> ( B C. A \/ B = A ) )
18		dfss2 3591	. . . . 5 |- ( B C_ A <-> A. y ( y e. B -> y e. A ) )
19	17, 18	bitr3i 266	. . . 4 |- ( ( B C. A \/ B = A ) <-> A. y ( y e. B -> y e. A ) )
20	13, 16, 19	3imtr4g 285	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( -. ( A C. B \/ A = B ) -> ( B C. A \/ B = A ) ) )
21	20	orrd 393	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( ( A C. B \/ A = B ) \/ ( B C. A \/ B = A ) ) )
22		df-3or 1038	. . 3 |- ( ( A C. B \/ A = B \/ B C. A ) <-> ( ( A C. B \/ A = B ) \/ B C. A ) )
23		or32 549	. . 3 |- ( ( ( A C. B \/ A = B ) \/ B C. A ) <-> ( ( A C. B \/ B C. A ) \/ A = B ) )
24		orordir 553	. . . 4 |- ( ( ( A C. B \/ B C. A ) \/ A = B ) <-> ( ( A C. B \/ A = B ) \/ ( B C. A \/ A = B ) ) )
25		eqcom 2629	. . . . . 6 |- ( B = A <-> A = B )
26	25	orbi2i 541	. . . . 5 |- ( ( B C. A \/ B = A ) <-> ( B C. A \/ A = B ) )
27	26	orbi2i 541	. . . 4 |- ( ( ( A C. B \/ A = B ) \/ ( B C. A \/ B = A ) ) <-> ( ( A C. B \/ A = B ) \/ ( B C. A \/ A = B ) ) )
28	24, 27	bitr4i 267	. . 3 |- ( ( ( A C. B \/ B C. A ) \/ A = B ) <-> ( ( A C. B \/ A = B ) \/ ( B C. A \/ B = A ) ) )
29	22, 23, 28	3bitri 286	. 2 |- ( ( A C. B \/ A = B \/ B C. A ) <-> ( ( A C. B \/ A = B ) \/ ( B C. A \/ B = A ) ) )
30	21, 29	sylibr 224	1 |- ( ( A e. P. /\ B e. P. ) -> ( A C. B \/ A = B \/ B C. A ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   \/ w3o 1036   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   C_ wss 3574   C. wpss 3575   class class class wbr 4653   Q.cnq 9674   <Q cltq 9680   P.cnp 9681

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-omul 7565  df-er 7742  df-ni 9694  df-mi 9696  df-lti 9697  df-ltpq 9732  df-enq 9733  df-nq 9734  df-ltnq 9740  df-np 9803

This theorem is referenced by:  ltsopr  9854