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Theorem supxrunb2 11384
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 3448 . . . . . . . 8  |-  ( A 
C_  RR*  ->  ( z  e.  A  ->  z  e. 
RR* ) )
2 pnfnlt 11209 . . . . . . . 8  |-  ( z  e.  RR*  ->  -. +oo  <  z )
31, 2syl6 33 . . . . . . 7  |-  ( A 
C_  RR*  ->  ( z  e.  A  ->  -. +oo  <  z ) )
43ralrimiv 2820 . . . . . 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 4393 . . . . . . . . . . . . . . 15  |-  ( x  =  z  ->  (
x  <  y  <->  z  <  y ) )
76rexbidv 2844 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  ( E. y  e.  A  x  <  y  <->  E. y  e.  A  z  <  y ) )
87rspcva 3167 . . . . . . . . . . . . 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 2820 . . . . 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 11193 . . . . 5  |- +oo  e.  RR*
19 supxr 11376 . . . . 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 9530 . . . . . . 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 11203 . . . . . . . . 9  |-  ( x  e.  RR  ->  x  < +oo )
26 breq2 4394 . . . . . . . . 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 11219 . . . . . . . 8  |-  <  Or  RR*
3130a1i 11 . . . . . . 7  |-  ( ( ( A  C_  RR*  /\  x  e.  RR )  /\  sup ( A ,  RR* ,  <  )  = +oo )  ->  <  Or  RR* )
32 xrsupss 11372 . . . . . . . 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 7811 . . . . . 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 2901 . 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 1370    e. wcel 1758   A.wral 2795   E.wrex 2796    C_ wss 3426   class class class wbr 4390    Or wor 4738   supcsup 7791   RRcr 9382   +oocpnf 9516   RR*cxr 9518    < clt 9519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-po 4739  df-so 4740  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-sup 7792  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699
This theorem is referenced by:  supxrbnd2  11386  supxrbnd  11392
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