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Theorem psslinpr 9409
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.
StepHypRef Expression
1 elprnq 9369 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  x  e.  Q. )
2 prub 9372 . . . . . . . . . . . . 13  |-  ( ( ( B  e.  P.  /\  y  e.  B )  /\  x  e.  Q. )  ->  ( -.  x  e.  B  ->  y  <Q  x ) )
31, 2sylan2 474 . . . . . . . . . . . 12  |-  ( ( ( B  e.  P.  /\  y  e.  B )  /\  ( A  e. 
P.  /\  x  e.  A ) )  -> 
( -.  x  e.  B  ->  y  <Q  x ) )
4 prcdnq 9371 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  ( y  <Q  x  ->  y  e.  A ) )
54adantl 466 . . . . . . . . . . . 12  |-  ( ( ( B  e.  P.  /\  y  e.  B )  /\  ( A  e. 
P.  /\  x  e.  A ) )  -> 
( y  <Q  x  ->  y  e.  A ) )
63, 5syld 44 . . . . . . . . . . 11  |-  ( ( ( B  e.  P.  /\  y  e.  B )  /\  ( A  e. 
P.  /\  x  e.  A ) )  -> 
( -.  x  e.  B  ->  y  e.  A ) )
76exp43 612 . . . . . . . . . 10  |-  ( B  e.  P.  ->  (
y  e.  B  -> 
( A  e.  P.  ->  ( x  e.  A  ->  ( -.  x  e.  B  ->  y  e.  A ) ) ) ) )
87com3r 79 . . . . . . . . 9  |-  ( A  e.  P.  ->  ( B  e.  P.  ->  ( y  e.  B  -> 
( x  e.  A  ->  ( -.  x  e.  B  ->  y  e.  A ) ) ) ) )
98imp 429 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( y  e.  B  ->  ( x  e.  A  ->  ( -.  x  e.  B  ->  y  e.  A ) ) ) )
109imp4a 589 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( y  e.  B  ->  ( ( x  e.  A  /\  -.  x  e.  B )  ->  y  e.  A ) ) )
1110com23 78 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( x  e.  A  /\  -.  x  e.  B )  ->  (
y  e.  B  -> 
y  e.  A ) ) )
1211alrimdv 1697 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( x  e.  A  /\  -.  x  e.  B )  ->  A. y
( y  e.  B  ->  y  e.  A ) ) )
1312exlimdv 1700 . . . 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 3562 . . . . 5  |-  ( -.  A  C_  B  <->  E. x
( x  e.  A  /\  -.  x  e.  B
) )
15 sspss 3603 . . . . 5  |-  ( A 
C_  B  <->  ( A  C.  B  \/  A  =  B ) )
1614, 15xchnxbi 308 . . . 4  |-  ( -.  ( A  C.  B  \/  A  =  B
)  <->  E. x ( x  e.  A  /\  -.  x  e.  B )
)
17 sspss 3603 . . . . 5  |-  ( B 
C_  A  <->  ( B  C.  A  \/  B  =  A ) )
18 dfss2 3493 . . . . 5  |-  ( B 
C_  A  <->  A. y
( y  e.  B  ->  y  e.  A ) )
1917, 18bitr3i 251 . . . 4  |-  ( ( B  C.  A  \/  B  =  A )  <->  A. y ( y  e.  B  ->  y  e.  A ) )
2013, 16, 193imtr4g 270 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( -.  ( A 
C.  B  \/  A  =  B )  ->  ( B  C.  A  \/  B  =  A ) ) )
2120orrd 378 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  C.  B  \/  A  =  B )  \/  ( B  C.  A  \/  B  =  A ) ) )
22 df-3or 974 . . 3  |-  ( ( A  C.  B  \/  A  =  B  \/  B  C.  A )  <->  ( ( A  C.  B  \/  A  =  B )  \/  B  C.  A ) )
23 or32 527 . . 3  |-  ( ( ( A  C.  B  \/  A  =  B
)  \/  B  C.  A )  <->  ( ( A  C.  B  \/  B  C.  A )  \/  A  =  B ) )
24 orordir 531 . . . 4  |-  ( ( ( A  C.  B  \/  B  C.  A )  \/  A  =  B )  <->  ( ( A 
C.  B  \/  A  =  B )  \/  ( B  C.  A  \/  A  =  B ) ) )
25 eqcom 2476 . . . . . 6  |-  ( B  =  A  <->  A  =  B )
2625orbi2i 519 . . . . 5  |-  ( ( B  C.  A  \/  B  =  A )  <->  ( B  C.  A  \/  A  =  B )
)
2726orbi2i 519 . . . 4  |-  ( ( ( A  C.  B  \/  A  =  B
)  \/  ( B 
C.  A  \/  B  =  A ) )  <->  ( ( A  C.  B  \/  A  =  B )  \/  ( B  C.  A  \/  A  =  B ) ) )
2824, 27bitr4i 252 . . 3  |-  ( ( ( A  C.  B  \/  B  C.  A )  \/  A  =  B )  <->  ( ( A 
C.  B  \/  A  =  B )  \/  ( B  C.  A  \/  B  =  A ) ) )
2922, 23, 283bitri 271 . 2  |-  ( ( A  C.  B  \/  A  =  B  \/  B  C.  A )  <->  ( ( A  C.  B  \/  A  =  B )  \/  ( B  C.  A  \/  B  =  A ) ) )
3021, 29sylibr 212 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  C.  B  \/  A  =  B  \/  B  C.  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    \/ w3o 972   A.wal 1377    = wceq 1379   E.wex 1596    e. wcel 1767    C_ wss 3476    C. wpss 3477   class class class wbr 4447   Q.cnq 9230    <Q cltq 9236   P.cnp 9237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-oadd 7134  df-omul 7135  df-er 7311  df-ni 9250  df-mi 9252  df-lti 9253  df-ltpq 9288  df-enq 9289  df-nq 9290  df-ltnq 9296  df-np 9359
This theorem is referenced by:  ltsopr  9410
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