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Theorem psslinpr 9402
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 9362 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  x  e.  Q. )
2 prub 9365 . . . . . . . . . . . . 13  |-  ( ( ( B  e.  P.  /\  y  e.  B )  /\  x  e.  Q. )  ->  ( -.  x  e.  B  ->  y  <Q  x ) )
31, 2sylan2 476 . . . . . . . . . . . 12  |-  ( ( ( B  e.  P.  /\  y  e.  B )  /\  ( A  e. 
P.  /\  x  e.  A ) )  -> 
( -.  x  e.  B  ->  y  <Q  x ) )
4 prcdnq 9364 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  ( y  <Q  x  ->  y  e.  A ) )
54adantl 467 . . . . . . . . . . . 12  |-  ( ( ( B  e.  P.  /\  y  e.  B )  /\  ( A  e. 
P.  /\  x  e.  A ) )  -> 
( y  <Q  x  ->  y  e.  A ) )
63, 5syld 45 . . . . . . . . . . 11  |-  ( ( ( B  e.  P.  /\  y  e.  B )  /\  ( A  e. 
P.  /\  x  e.  A ) )  -> 
( -.  x  e.  B  ->  y  e.  A ) )
76exp43 615 . . . . . . . . . 10  |-  ( B  e.  P.  ->  (
y  e.  B  -> 
( A  e.  P.  ->  ( x  e.  A  ->  ( -.  x  e.  B  ->  y  e.  A ) ) ) ) )
87com3r 82 . . . . . . . . 9  |-  ( A  e.  P.  ->  ( B  e.  P.  ->  ( y  e.  B  -> 
( x  e.  A  ->  ( -.  x  e.  B  ->  y  e.  A ) ) ) ) )
98imp 430 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( y  e.  B  ->  ( x  e.  A  ->  ( -.  x  e.  B  ->  y  e.  A ) ) ) )
109imp4a 592 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( y  e.  B  ->  ( ( x  e.  A  /\  -.  x  e.  B )  ->  y  e.  A ) ) )
1110com23 81 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( x  e.  A  /\  -.  x  e.  B )  ->  (
y  e.  B  -> 
y  e.  A ) ) )
1211alrimdv 1769 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( x  e.  A  /\  -.  x  e.  B )  ->  A. y
( y  e.  B  ->  y  e.  A ) ) )
1312exlimdv 1772 . . . 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 3460 . . . . 5  |-  ( -.  A  C_  B  <->  E. x
( x  e.  A  /\  -.  x  e.  B
) )
15 sspss 3502 . . . . 5  |-  ( A 
C_  B  <->  ( A  C.  B  \/  A  =  B ) )
1614, 15xchnxbi 309 . . . 4  |-  ( -.  ( A  C.  B  \/  A  =  B
)  <->  E. x ( x  e.  A  /\  -.  x  e.  B )
)
17 sspss 3502 . . . . 5  |-  ( B 
C_  A  <->  ( B  C.  A  \/  B  =  A ) )
18 dfss2 3391 . . . . 5  |-  ( B 
C_  A  <->  A. y
( y  e.  B  ->  y  e.  A ) )
1917, 18bitr3i 254 . . . 4  |-  ( ( B  C.  A  \/  B  =  A )  <->  A. y ( y  e.  B  ->  y  e.  A ) )
2013, 16, 193imtr4g 273 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( -.  ( A 
C.  B  \/  A  =  B )  ->  ( B  C.  A  \/  B  =  A ) ) )
2120orrd 379 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  C.  B  \/  A  =  B )  \/  ( B  C.  A  \/  B  =  A ) ) )
22 df-3or 983 . . 3  |-  ( ( A  C.  B  \/  A  =  B  \/  B  C.  A )  <->  ( ( A  C.  B  \/  A  =  B )  \/  B  C.  A ) )
23 or32 529 . . 3  |-  ( ( ( A  C.  B  \/  A  =  B
)  \/  B  C.  A )  <->  ( ( A  C.  B  \/  B  C.  A )  \/  A  =  B ) )
24 orordir 533 . . . 4  |-  ( ( ( A  C.  B  \/  B  C.  A )  \/  A  =  B )  <->  ( ( A 
C.  B  \/  A  =  B )  \/  ( B  C.  A  \/  A  =  B ) ) )
25 eqcom 2430 . . . . . 6  |-  ( B  =  A  <->  A  =  B )
2625orbi2i 521 . . . . 5  |-  ( ( B  C.  A  \/  B  =  A )  <->  ( B  C.  A  \/  A  =  B )
)
2726orbi2i 521 . . . 4  |-  ( ( ( A  C.  B  \/  A  =  B
)  \/  ( B 
C.  A  \/  B  =  A ) )  <->  ( ( A  C.  B  \/  A  =  B )  \/  ( B  C.  A  \/  A  =  B ) ) )
2824, 27bitr4i 255 . . 3  |-  ( ( ( A  C.  B  \/  B  C.  A )  \/  A  =  B )  <->  ( ( A 
C.  B  \/  A  =  B )  \/  ( B  C.  A  \/  B  =  A ) ) )
2922, 23, 283bitri 274 . 2  |-  ( ( A  C.  B  \/  A  =  B  \/  B  C.  A )  <->  ( ( A  C.  B  \/  A  =  B )  \/  ( B  C.  A  \/  B  =  A ) ) )
3021, 29sylibr 215 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 369    /\ wa 370    \/ w3o 981   A.wal 1435    = wceq 1437   E.wex 1657    e. wcel 1872    C_ wss 3374    C. wpss 3375   class class class wbr 4361   Q.cnq 9223    <Q cltq 9229   P.cnp 9230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-pss 3390  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4158  df-iun 4239  df-br 4362  df-opab 4421  df-mpt 4422  df-tr 4457  df-eprel 4702  df-id 4706  df-po 4712  df-so 4713  df-fr 4750  df-we 4752  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-pred 5337  df-ord 5383  df-on 5384  df-lim 5385  df-suc 5386  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-ov 6247  df-oprab 6248  df-mpt2 6249  df-om 6646  df-1st 6746  df-2nd 6747  df-wrecs 6978  df-recs 7040  df-rdg 7078  df-oadd 7136  df-omul 7137  df-er 7313  df-ni 9243  df-mi 9245  df-lti 9246  df-ltpq 9281  df-enq 9282  df-nq 9283  df-ltnq 9289  df-np 9352
This theorem is referenced by:  ltsopr  9403
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