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Theorem psslinpr 8650
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 ) )
Dummy variables  x  y are mutually distinct and distinct from all other variables.

Proof of Theorem psslinpr
StepHypRef Expression
1 elprnq 8610 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  x  e.  Q. )
2 prub 8613 . . . . . . . . . . . . 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 8612 . . . . . . . . . . . . 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 597 . . . . . . . . . 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 574 . . . . . . 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 1620 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( x  e.  A  /\  -.  x  e.  B )  ->  A. y
( y  e.  B  ->  y  e.  A ) ) )
1312exlimdv 1665 . . . 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 3237 . . . . 5  |-  ( -.  A  C_  B  <->  E. x
( x  e.  A  /\  -.  x  e.  B
) )
15 sspss 3276 . . . . 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 3276 . . . . 5  |-  ( B 
C_  A  <->  ( B  C.  A  \/  B  =  A ) )
18 dfss2 3170 . . . . 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 937 . . 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 2286 . . . . . 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 935   A.wal 1528   E.wex 1529    = wceq 1624    e. wcel 1685    C_ wss 3153    C. wpss 3154   class class class wbr 4024   Q.cnq 8469    <Q cltq 8475   P.cnp 8476
This theorem is referenced by:  ltsopr  8651
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-recs 6383  df-rdg 6418  df-oadd 6478  df-omul 6479  df-er 6655  df-ni 8491  df-mi 8493  df-lti 8494  df-ltpq 8529  df-enq 8530  df-nq 8531  df-ltnq 8537  df-np 8600
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