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Theorem supxrunb2 11483
Description: The supremum of an unbounded-above set of extended reals is plus infinity. (Contributed by NM, 19-Jan-2006.)
Assertion
Ref Expression
supxrunb2  |-  ( A 
C_  RR*  ->  ( A. x  e.  RR  E. y  e.  A  x  <  y  <->  sup ( A ,  RR* ,  <  )  = +oo ) )
Distinct variable group:    x, y, A

Proof of Theorem supxrunb2
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3435 . . . . . . . 8  |-  ( A 
C_  RR*  ->  ( z  e.  A  ->  z  e. 
RR* ) )
2 pnfnlt 11308 . . . . . . . 8  |-  ( z  e.  RR*  ->  -. +oo  <  z )
31, 2syl6 31 . . . . . . 7  |-  ( A 
C_  RR*  ->  ( z  e.  A  ->  -. +oo  <  z ) )
43ralrimiv 2815 . . . . . 6  |-  ( A 
C_  RR*  ->  A. z  e.  A  -. +oo  <  z )
54adantr 463 . . . . 5  |-  ( ( A  C_  RR*  /\  A. x  e.  RR  E. y  e.  A  x  <  y )  ->  A. z  e.  A  -. +oo  <  z )
6 breq1 4397 . . . . . . . . . . . . . . 15  |-  ( x  =  z  ->  (
x  <  y  <->  z  <  y ) )
76rexbidv 2917 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  ( E. y  e.  A  x  <  y  <->  E. y  e.  A  z  <  y ) )
87rspcva 3157 . . . . . . . . . . . . 13  |-  ( ( z  e.  RR  /\  A. x  e.  RR  E. y  e.  A  x  <  y )  ->  E. y  e.  A  z  <  y )
98adantrr 715 . . . . . . . . . . . 12  |-  ( ( z  e.  RR  /\  ( A. x  e.  RR  E. y  e.  A  x  <  y  /\  A  C_ 
RR* ) )  ->  E. y  e.  A  z  <  y )
109ancoms 451 . . . . . . . . . . 11  |-  ( ( ( A. x  e.  RR  E. y  e.  A  x  <  y  /\  A  C_  RR* )  /\  z  e.  RR )  ->  E. y  e.  A  z  <  y )
1110exp31 602 . . . . . . . . . 10  |-  ( A. x  e.  RR  E. y  e.  A  x  <  y  ->  ( A  C_  RR* 
->  ( z  e.  RR  ->  E. y  e.  A  z  <  y ) ) )
1211a1dd 44 . . . . . . . . 9  |-  ( A. x  e.  RR  E. y  e.  A  x  <  y  ->  ( A  C_  RR* 
->  ( z  < +oo  ->  ( z  e.  RR  ->  E. y  e.  A  z  <  y ) ) ) )
1312com4r 86 . . . . . . . 8  |-  ( z  e.  RR  ->  ( A. x  e.  RR  E. y  e.  A  x  <  y  ->  ( A  C_  RR*  ->  ( z  < +oo  ->  E. y  e.  A  z  <  y ) ) ) )
1413com13 80 . . . . . . 7  |-  ( A 
C_  RR*  ->  ( A. x  e.  RR  E. y  e.  A  x  <  y  ->  ( z  e.  RR  ->  ( z  < +oo  ->  E. y  e.  A  z  <  y ) ) ) )
1514imp 427 . . . . . 6  |-  ( ( A  C_  RR*  /\  A. x  e.  RR  E. y  e.  A  x  <  y )  ->  ( z  e.  RR  ->  ( z  < +oo  ->  E. y  e.  A  z  <  y ) ) )
1615ralrimiv 2815 . . . . 5  |-  ( ( A  C_  RR*  /\  A. x  e.  RR  E. y  e.  A  x  <  y )  ->  A. z  e.  RR  ( z  < +oo  ->  E. y  e.  A  z  <  y ) )
175, 16jca 530 . . . 4  |-  ( ( A  C_  RR*  /\  A. x  e.  RR  E. y  e.  A  x  <  y )  ->  ( A. z  e.  A  -. +oo 
<  z  /\  A. z  e.  RR  ( z  < +oo  ->  E. y  e.  A  z  <  y ) ) )
18 pnfxr 11292 . . . . 5  |- +oo  e.  RR*
19 supxr 11475 . . . . 5  |-  ( ( ( A  C_  RR*  /\ +oo  e.  RR* )  /\  ( A. z  e.  A  -. +oo  <  z  /\  A. z  e.  RR  (
z  < +oo  ->  E. y  e.  A  z  <  y ) ) )  ->  sup ( A ,  RR* ,  <  )  = +oo )
2018, 19mpanl2 679 . . . 4  |-  ( ( A  C_  RR*  /\  ( A. z  e.  A  -. +oo  <  z  /\  A. z  e.  RR  (
z  < +oo  ->  E. y  e.  A  z  <  y ) ) )  ->  sup ( A ,  RR* ,  <  )  = +oo )
2117, 20syldan 468 . . 3  |-  ( ( A  C_  RR*  /\  A. x  e.  RR  E. y  e.  A  x  <  y )  ->  sup ( A ,  RR* ,  <  )  = +oo )
2221ex 432 . 2  |-  ( A 
C_  RR*  ->  ( A. x  e.  RR  E. y  e.  A  x  <  y  ->  sup ( A ,  RR* ,  <  )  = +oo ) )
23 rexr 9589 . . . . . . 7  |-  ( x  e.  RR  ->  x  e.  RR* )
2423ad2antlr 725 . . . . . 6  |-  ( ( ( A  C_  RR*  /\  x  e.  RR )  /\  sup ( A ,  RR* ,  <  )  = +oo )  ->  x  e.  RR* )
25 ltpnf 11302 . . . . . . . . 9  |-  ( x  e.  RR  ->  x  < +oo )
26 breq2 4398 . . . . . . . . 9  |-  ( sup ( A ,  RR* ,  <  )  = +oo  ->  ( x  <  sup ( A ,  RR* ,  <  )  <-> 
x  < +oo )
)
2725, 26syl5ibr 221 . . . . . . . 8  |-  ( sup ( A ,  RR* ,  <  )  = +oo  ->  ( x  e.  RR  ->  x  <  sup ( A ,  RR* ,  <  ) ) )
2827impcom 428 . . . . . . 7  |-  ( ( x  e.  RR  /\  sup ( A ,  RR* ,  <  )  = +oo )  ->  x  <  sup ( A ,  RR* ,  <  ) )
2928adantll 712 . . . . . 6  |-  ( ( ( A  C_  RR*  /\  x  e.  RR )  /\  sup ( A ,  RR* ,  <  )  = +oo )  ->  x  <  sup ( A ,  RR* ,  <  ) )
30 xrltso 11318 . . . . . . . 8  |-  <  Or  RR*
3130a1i 11 . . . . . . 7  |-  ( ( ( A  C_  RR*  /\  x  e.  RR )  /\  sup ( A ,  RR* ,  <  )  = +oo )  ->  <  Or  RR* )
32 xrsupss 11471 . . . . . . . 8  |-  ( A 
C_  RR*  ->  E. z  e.  RR*  ( A. w  e.  A  -.  z  <  w  /\  A. w  e.  RR*  ( w  < 
z  ->  E. y  e.  A  w  <  y ) ) )
3332ad2antrr 724 . . . . . . 7  |-  ( ( ( A  C_  RR*  /\  x  e.  RR )  /\  sup ( A ,  RR* ,  <  )  = +oo )  ->  E. z  e.  RR*  ( A. w  e.  A  -.  z  <  w  /\  A. w  e.  RR*  (
w  <  z  ->  E. y  e.  A  w  <  y ) ) )
3431, 33suplub 7873 . . . . . 6  |-  ( ( ( A  C_  RR*  /\  x  e.  RR )  /\  sup ( A ,  RR* ,  <  )  = +oo )  -> 
( ( x  e. 
RR*  /\  x  <  sup ( A ,  RR* ,  <  ) )  ->  E. y  e.  A  x  <  y ) )
3524, 29, 34mp2and 677 . . . . 5  |-  ( ( ( A  C_  RR*  /\  x  e.  RR )  /\  sup ( A ,  RR* ,  <  )  = +oo )  ->  E. y  e.  A  x  <  y )
3635exp31 602 . . . 4  |-  ( A 
C_  RR*  ->  ( x  e.  RR  ->  ( sup ( A ,  RR* ,  <  )  = +oo  ->  E. y  e.  A  x  <  y ) ) )
3736com23 78 . . 3  |-  ( A 
C_  RR*  ->  ( sup ( A ,  RR* ,  <  )  = +oo  ->  (
x  e.  RR  ->  E. y  e.  A  x  <  y ) ) )
3837ralrimdv 2819 . 2  |-  ( A 
C_  RR*  ->  ( sup ( A ,  RR* ,  <  )  = +oo  ->  A. x  e.  RR  E. y  e.  A  x  <  y
) )
3922, 38impbid 191 1  |-  ( A 
C_  RR*  ->  ( A. x  e.  RR  E. y  e.  A  x  <  y  <->  sup ( A ,  RR* ,  <  )  = +oo ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2753   E.wrex 2754    C_ wss 3413   class class class wbr 4394    Or wor 4742   supcsup 7854   RRcr 9441   +oocpnf 9575   RR*cxr 9577    < clt 9578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519  ax-pre-sup 9520
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-po 4743  df-so 4744  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-er 7268  df-en 7475  df-dom 7476  df-sdom 7477  df-sup 7855  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764
This theorem is referenced by:  supxrbnd2  11485  supxrbnd  11491
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