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Theorem psslinpr 8623
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
StepHypRef Expression
1 elprnq 8583 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  x  e.  Q. )
2 prub 8586 . . . . . . . . . . . . 13  |-  ( ( ( B  e.  P.  /\  y  e.  B )  /\  x  e.  Q. )  ->  ( -.  x  e.  B  ->  y  <Q  x ) )
31, 2sylan2 462 . . . . . . . . . . . 12  |-  ( ( ( B  e.  P.  /\  y  e.  B )  /\  ( A  e. 
P.  /\  x  e.  A ) )  -> 
( -.  x  e.  B  ->  y  <Q  x ) )
4 prcdnq 8585 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  ( y  <Q  x  ->  y  e.  A ) )
54adantl 454 . . . . . . . . . . . 12  |-  ( ( ( B  e.  P.  /\  y  e.  B )  /\  ( A  e. 
P.  /\  x  e.  A ) )  -> 
( y  <Q  x  ->  y  e.  A ) )
63, 5syld 42 . . . . . . . . . . 11  |-  ( ( ( B  e.  P.  /\  y  e.  B )  /\  ( A  e. 
P.  /\  x  e.  A ) )  -> 
( -.  x  e.  B  ->  y  e.  A ) )
76exp43 598 . . . . . . . . . 10  |-  ( B  e.  P.  ->  (
y  e.  B  -> 
( A  e.  P.  ->  ( x  e.  A  ->  ( -.  x  e.  B  ->  y  e.  A ) ) ) ) )
87com3r 75 . . . . . . . . 9  |-  ( A  e.  P.  ->  ( B  e.  P.  ->  ( y  e.  B  -> 
( x  e.  A  ->  ( -.  x  e.  B  ->  y  e.  A ) ) ) ) )
98imp 420 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( y  e.  B  ->  ( x  e.  A  ->  ( -.  x  e.  B  ->  y  e.  A ) ) ) )
109imp4a 575 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( y  e.  B  ->  ( ( x  e.  A  /\  -.  x  e.  B )  ->  y  e.  A ) ) )
1110com23 74 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( x  e.  A  /\  -.  x  e.  B )  ->  (
y  e.  B  -> 
y  e.  A ) ) )
1211alrimdv 2015 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( x  e.  A  /\  -.  x  e.  B )  ->  A. y
( y  e.  B  ->  y  e.  A ) ) )
1312exlimdv 1933 . . . 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 3211 . . . . 5  |-  ( -.  A  C_  B  <->  E. x
( x  e.  A  /\  -.  x  e.  B
) )
15 sspss 3250 . . . . 5  |-  ( A 
C_  B  <->  ( A  C.  B  \/  A  =  B ) )
1614, 15xchnxbi 301 . . . 4  |-  ( -.  ( A  C.  B  \/  A  =  B
)  <->  E. x ( x  e.  A  /\  -.  x  e.  B )
)
17 sspss 3250 . . . . 5  |-  ( B 
C_  A  <->  ( B  C.  A  \/  B  =  A ) )
18 dfss2 3144 . . . . 5  |-  ( B 
C_  A  <->  A. y
( y  e.  B  ->  y  e.  A ) )
1917, 18bitr3i 244 . . . 4  |-  ( ( B  C.  A  \/  B  =  A )  <->  A. y ( y  e.  B  ->  y  e.  A ) )
2013, 16, 193imtr4g 263 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( -.  ( A 
C.  B  \/  A  =  B )  ->  ( B  C.  A  \/  B  =  A ) ) )
2120orrd 369 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  C.  B  \/  A  =  B )  \/  ( B  C.  A  \/  B  =  A ) ) )
22 df-3or 940 . . 3  |-  ( ( A  C.  B  \/  A  =  B  \/  B  C.  A )  <->  ( ( A  C.  B  \/  A  =  B )  \/  B  C.  A ) )
23 or32 515 . . 3  |-  ( ( ( A  C.  B  \/  A  =  B
)  \/  B  C.  A )  <->  ( ( A  C.  B  \/  B  C.  A )  \/  A  =  B ) )
24 orordir 519 . . . 4  |-  ( ( ( A  C.  B  \/  B  C.  A )  \/  A  =  B )  <->  ( ( A 
C.  B  \/  A  =  B )  \/  ( B  C.  A  \/  A  =  B ) ) )
25 eqcom 2260 . . . . . 6  |-  ( B  =  A  <->  A  =  B )
2625orbi2i 507 . . . . 5  |-  ( ( B  C.  A  \/  B  =  A )  <->  ( B  C.  A  \/  A  =  B )
)
2726orbi2i 507 . . . 4  |-  ( ( ( A  C.  B  \/  A  =  B
)  \/  ( B 
C.  A  \/  B  =  A ) )  <->  ( ( A  C.  B  \/  A  =  B )  \/  ( B  C.  A  \/  A  =  B ) ) )
2824, 27bitr4i 245 . . 3  |-  ( ( ( A  C.  B  \/  B  C.  A )  \/  A  =  B )  <->  ( ( A 
C.  B  \/  A  =  B )  \/  ( B  C.  A  \/  B  =  A ) ) )
2922, 23, 283bitri 264 . 2  |-  ( ( A  C.  B  \/  A  =  B  \/  B  C.  A )  <->  ( ( A  C.  B  \/  A  =  B )  \/  ( B  C.  A  \/  B  =  A ) ) )
3021, 29sylibr 205 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  C.  B  \/  A  =  B  \/  B  C.  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    \/ wo 359    /\ wa 360    \/ w3o 938   A.wal 1532   E.wex 1537    = wceq 1619    e. wcel 1621    C_ wss 3127    C. wpss 3128   class class class wbr 3997   Q.cnq 8442    <Q cltq 8448   P.cnp 8449
This theorem is referenced by:  ltsopr  8624
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-recs 6356  df-rdg 6391  df-oadd 6451  df-omul 6452  df-er 6628  df-ni 8464  df-mi 8466  df-lti 8467  df-ltpq 8502  df-enq 8503  df-nq 8504  df-ltnq 8510  df-np 8573
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