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Theorem psslinpr 8655
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 8615 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  x  e.  Q. )
2 prub 8618 . . . . . . . . . . . . 13  |-  ( ( ( B  e.  P.  /\  y  e.  B )  /\  x  e.  Q. )  ->  ( -.  x  e.  B  ->  y  <Q  x ) )
31, 2sylan2 460 . . . . . . . . . . . 12  |-  ( ( ( B  e.  P.  /\  y  e.  B )  /\  ( A  e. 
P.  /\  x  e.  A ) )  -> 
( -.  x  e.  B  ->  y  <Q  x ) )
4 prcdnq 8617 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  ( y  <Q  x  ->  y  e.  A ) )
54adantl 452 . . . . . . . . . . . 12  |-  ( ( ( B  e.  P.  /\  y  e.  B )  /\  ( A  e. 
P.  /\  x  e.  A ) )  -> 
( y  <Q  x  ->  y  e.  A ) )
63, 5syld 40 . . . . . . . . . . 11  |-  ( ( ( B  e.  P.  /\  y  e.  B )  /\  ( A  e. 
P.  /\  x  e.  A ) )  -> 
( -.  x  e.  B  ->  y  e.  A ) )
76exp43 595 . . . . . . . . . 10  |-  ( B  e.  P.  ->  (
y  e.  B  -> 
( A  e.  P.  ->  ( x  e.  A  ->  ( -.  x  e.  B  ->  y  e.  A ) ) ) ) )
87com3r 73 . . . . . . . . 9  |-  ( A  e.  P.  ->  ( B  e.  P.  ->  ( y  e.  B  -> 
( x  e.  A  ->  ( -.  x  e.  B  ->  y  e.  A ) ) ) ) )
98imp 418 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( y  e.  B  ->  ( x  e.  A  ->  ( -.  x  e.  B  ->  y  e.  A ) ) ) )
109imp4a 572 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( y  e.  B  ->  ( ( x  e.  A  /\  -.  x  e.  B )  ->  y  e.  A ) ) )
1110com23 72 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( x  e.  A  /\  -.  x  e.  B )  ->  (
y  e.  B  -> 
y  e.  A ) ) )
1211alrimdv 1619 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( x  e.  A  /\  -.  x  e.  B )  ->  A. y
( y  e.  B  ->  y  e.  A ) ) )
1312exlimdv 1664 . . . 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 3236 . . . . 5  |-  ( -.  A  C_  B  <->  E. x
( x  e.  A  /\  -.  x  e.  B
) )
15 sspss 3275 . . . . 5  |-  ( A 
C_  B  <->  ( A  C.  B  \/  A  =  B ) )
1614, 15xchnxbi 299 . . . 4  |-  ( -.  ( A  C.  B  \/  A  =  B
)  <->  E. x ( x  e.  A  /\  -.  x  e.  B )
)
17 sspss 3275 . . . . 5  |-  ( B 
C_  A  <->  ( B  C.  A  \/  B  =  A ) )
18 dfss2 3169 . . . . 5  |-  ( B 
C_  A  <->  A. y
( y  e.  B  ->  y  e.  A ) )
1917, 18bitr3i 242 . . . 4  |-  ( ( B  C.  A  \/  B  =  A )  <->  A. y ( y  e.  B  ->  y  e.  A ) )
2013, 16, 193imtr4g 261 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( -.  ( A 
C.  B  \/  A  =  B )  ->  ( B  C.  A  \/  B  =  A ) ) )
2120orrd 367 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  C.  B  \/  A  =  B )  \/  ( B  C.  A  \/  B  =  A ) ) )
22 df-3or 935 . . 3  |-  ( ( A  C.  B  \/  A  =  B  \/  B  C.  A )  <->  ( ( A  C.  B  \/  A  =  B )  \/  B  C.  A ) )
23 or32 513 . . 3  |-  ( ( ( A  C.  B  \/  A  =  B
)  \/  B  C.  A )  <->  ( ( A  C.  B  \/  B  C.  A )  \/  A  =  B ) )
24 orordir 517 . . . 4  |-  ( ( ( A  C.  B  \/  B  C.  A )  \/  A  =  B )  <->  ( ( A 
C.  B  \/  A  =  B )  \/  ( B  C.  A  \/  A  =  B ) ) )
25 eqcom 2285 . . . . . 6  |-  ( B  =  A  <->  A  =  B )
2625orbi2i 505 . . . . 5  |-  ( ( B  C.  A  \/  B  =  A )  <->  ( B  C.  A  \/  A  =  B )
)
2726orbi2i 505 . . . 4  |-  ( ( ( A  C.  B  \/  A  =  B
)  \/  ( B 
C.  A  \/  B  =  A ) )  <->  ( ( A  C.  B  \/  A  =  B )  \/  ( B  C.  A  \/  A  =  B ) ) )
2824, 27bitr4i 243 . . 3  |-  ( ( ( A  C.  B  \/  B  C.  A )  \/  A  =  B )  <->  ( ( A 
C.  B  \/  A  =  B )  \/  ( B  C.  A  \/  B  =  A ) ) )
2922, 23, 283bitri 262 . 2  |-  ( ( A  C.  B  \/  A  =  B  \/  B  C.  A )  <->  ( ( A  C.  B  \/  A  =  B )  \/  ( B  C.  A  \/  B  =  A ) ) )
3021, 29sylibr 203 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  C.  B  \/  A  =  B  \/  B  C.  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    \/ w3o 933   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684    C_ wss 3152    C. wpss 3153   class class class wbr 4023   Q.cnq 8474    <Q cltq 8480   P.cnp 8481
This theorem is referenced by:  ltsopr  8656
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-oadd 6483  df-omul 6484  df-er 6660  df-ni 8496  df-mi 8498  df-lti 8499  df-ltpq 8534  df-enq 8535  df-nq 8536  df-ltnq 8542  df-np 8605
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