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Theorem supxrunb2 11503
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 3493 . . . . . . . 8  |-  ( A 
C_  RR*  ->  ( z  e.  A  ->  z  e. 
RR* ) )
2 pnfnlt 11328 . . . . . . . 8  |-  ( z  e.  RR*  ->  -. +oo  <  z )
31, 2syl6 33 . . . . . . 7  |-  ( A 
C_  RR*  ->  ( z  e.  A  ->  -. +oo  <  z ) )
43ralrimiv 2871 . . . . . 6  |-  ( A 
C_  RR*  ->  A. z  e.  A  -. +oo  <  z )
54adantr 465 . . . . 5  |-  ( ( A  C_  RR*  /\  A. x  e.  RR  E. y  e.  A  x  <  y )  ->  A. z  e.  A  -. +oo  <  z )
6 breq1 4445 . . . . . . . . . . . . . . 15  |-  ( x  =  z  ->  (
x  <  y  <->  z  <  y ) )
76rexbidv 2968 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  ( E. y  e.  A  x  <  y  <->  E. y  e.  A  z  <  y ) )
87rspcva 3207 . . . . . . . . . . . . 13  |-  ( ( z  e.  RR  /\  A. x  e.  RR  E. y  e.  A  x  <  y )  ->  E. y  e.  A  z  <  y )
98adantrr 716 . . . . . . . . . . . 12  |-  ( ( z  e.  RR  /\  ( A. x  e.  RR  E. y  e.  A  x  <  y  /\  A  C_ 
RR* ) )  ->  E. y  e.  A  z  <  y )
109ancoms 453 . . . . . . . . . . 11  |-  ( ( ( A. x  e.  RR  E. y  e.  A  x  <  y  /\  A  C_  RR* )  /\  z  e.  RR )  ->  E. y  e.  A  z  <  y )
1110exp31 604 . . . . . . . . . 10  |-  ( A. x  e.  RR  E. y  e.  A  x  <  y  ->  ( A  C_  RR* 
->  ( z  e.  RR  ->  E. y  e.  A  z  <  y ) ) )
1211a1dd 46 . . . . . . . . 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 429 . . . . . 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 2871 . . . . 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 532 . . . 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 11312 . . . . 5  |- +oo  e.  RR*
19 supxr 11495 . . . . 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 681 . . . 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 470 . . 3  |-  ( ( A  C_  RR*  /\  A. x  e.  RR  E. y  e.  A  x  <  y )  ->  sup ( A ,  RR* ,  <  )  = +oo )
2221ex 434 . 2  |-  ( A 
C_  RR*  ->  ( A. x  e.  RR  E. y  e.  A  x  <  y  ->  sup ( A ,  RR* ,  <  )  = +oo ) )
23 rexr 9630 . . . . . . 7  |-  ( x  e.  RR  ->  x  e.  RR* )
2423ad2antlr 726 . . . . . 6  |-  ( ( ( A  C_  RR*  /\  x  e.  RR )  /\  sup ( A ,  RR* ,  <  )  = +oo )  ->  x  e.  RR* )
25 ltpnf 11322 . . . . . . . . 9  |-  ( x  e.  RR  ->  x  < +oo )
26 breq2 4446 . . . . . . . . 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 430 . . . . . . 7  |-  ( ( x  e.  RR  /\  sup ( A ,  RR* ,  <  )  = +oo )  ->  x  <  sup ( A ,  RR* ,  <  ) )
2928adantll 713 . . . . . 6  |-  ( ( ( A  C_  RR*  /\  x  e.  RR )  /\  sup ( A ,  RR* ,  <  )  = +oo )  ->  x  <  sup ( A ,  RR* ,  <  ) )
30 xrltso 11338 . . . . . . . 8  |-  <  Or  RR*
3130a1i 11 . . . . . . 7  |-  ( ( ( A  C_  RR*  /\  x  e.  RR )  /\  sup ( A ,  RR* ,  <  )  = +oo )  ->  <  Or  RR* )
32 xrsupss 11491 . . . . . . . 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 725 . . . . . . 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 7911 . . . . . 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 679 . . . . 5  |-  ( ( ( A  C_  RR*  /\  x  e.  RR )  /\  sup ( A ,  RR* ,  <  )  = +oo )  ->  E. y  e.  A  x  <  y )
3635exp31 604 . . . 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 2875 . 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 369    = wceq 1374    e. wcel 1762   A.wral 2809   E.wrex 2810    C_ wss 3471   class class class wbr 4442    Or wor 4794   supcsup 7891   RRcr 9482   +oocpnf 9616   RR*cxr 9618    < clt 9619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-po 4795  df-so 4796  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-sup 7892  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799
This theorem is referenced by:  supxrbnd2  11505  supxrbnd  11511
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