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Theorem supxrunb2 11613
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 3428 . . . . . . . 8  |-  ( A 
C_  RR*  ->  ( z  e.  A  ->  z  e. 
RR* ) )
2 pnfnlt 11437 . . . . . . . 8  |-  ( z  e.  RR*  ->  -. +oo  <  z )
31, 2syl6 34 . . . . . . 7  |-  ( A 
C_  RR*  ->  ( z  e.  A  ->  -. +oo  <  z ) )
43ralrimiv 2802 . . . . . 6  |-  ( A 
C_  RR*  ->  A. z  e.  A  -. +oo  <  z )
54adantr 467 . . . . 5  |-  ( ( A  C_  RR*  /\  A. x  e.  RR  E. y  e.  A  x  <  y )  ->  A. z  e.  A  -. +oo  <  z )
6 breq1 4408 . . . . . . . . . . . . . . 15  |-  ( x  =  z  ->  (
x  <  y  <->  z  <  y ) )
76rexbidv 2903 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  ( E. y  e.  A  x  <  y  <->  E. y  e.  A  z  <  y ) )
87rspcva 3150 . . . . . . . . . . . . 13  |-  ( ( z  e.  RR  /\  A. x  e.  RR  E. y  e.  A  x  <  y )  ->  E. y  e.  A  z  <  y )
98adantrr 724 . . . . . . . . . . . 12  |-  ( ( z  e.  RR  /\  ( A. x  e.  RR  E. y  e.  A  x  <  y  /\  A  C_ 
RR* ) )  ->  E. y  e.  A  z  <  y )
109ancoms 455 . . . . . . . . . . 11  |-  ( ( ( A. x  e.  RR  E. y  e.  A  x  <  y  /\  A  C_  RR* )  /\  z  e.  RR )  ->  E. y  e.  A  z  <  y )
1110exp31 609 . . . . . . . . . 10  |-  ( A. x  e.  RR  E. y  e.  A  x  <  y  ->  ( A  C_  RR* 
->  ( z  e.  RR  ->  E. y  e.  A  z  <  y ) ) )
1211a1dd 47 . . . . . . . . 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 89 . . . . . . . 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 83 . . . . . . 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 431 . . . . . 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 2802 . . . . 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 535 . . . 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 11419 . . . . 5  |- +oo  e.  RR*
19 supxr 11605 . . . . 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 688 . . . 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 473 . . 3  |-  ( ( A  C_  RR*  /\  A. x  e.  RR  E. y  e.  A  x  <  y )  ->  sup ( A ,  RR* ,  <  )  = +oo )
2221ex 436 . 2  |-  ( A 
C_  RR*  ->  ( A. x  e.  RR  E. y  e.  A  x  <  y  ->  sup ( A ,  RR* ,  <  )  = +oo ) )
23 rexr 9691 . . . . . . 7  |-  ( x  e.  RR  ->  x  e.  RR* )
2423ad2antlr 734 . . . . . 6  |-  ( ( ( A  C_  RR*  /\  x  e.  RR )  /\  sup ( A ,  RR* ,  <  )  = +oo )  ->  x  e.  RR* )
25 ltpnf 11429 . . . . . . . . 9  |-  ( x  e.  RR  ->  x  < +oo )
26 breq2 4409 . . . . . . . . 9  |-  ( sup ( A ,  RR* ,  <  )  = +oo  ->  ( x  <  sup ( A ,  RR* ,  <  )  <-> 
x  < +oo )
)
2725, 26syl5ibr 225 . . . . . . . 8  |-  ( sup ( A ,  RR* ,  <  )  = +oo  ->  ( x  e.  RR  ->  x  <  sup ( A ,  RR* ,  <  ) ) )
2827impcom 432 . . . . . . 7  |-  ( ( x  e.  RR  /\  sup ( A ,  RR* ,  <  )  = +oo )  ->  x  <  sup ( A ,  RR* ,  <  ) )
2928adantll 721 . . . . . 6  |-  ( ( ( A  C_  RR*  /\  x  e.  RR )  /\  sup ( A ,  RR* ,  <  )  = +oo )  ->  x  <  sup ( A ,  RR* ,  <  ) )
30 xrltso 11447 . . . . . . . 8  |-  <  Or  RR*
3130a1i 11 . . . . . . 7  |-  ( ( ( A  C_  RR*  /\  x  e.  RR )  /\  sup ( A ,  RR* ,  <  )  = +oo )  ->  <  Or  RR* )
32 xrsupss 11601 . . . . . . . 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 733 . . . . . . 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 7979 . . . . . 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 686 . . . . 5  |-  ( ( ( A  C_  RR*  /\  x  e.  RR )  /\  sup ( A ,  RR* ,  <  )  = +oo )  ->  E. y  e.  A  x  <  y )
3635exp31 609 . . . 4  |-  ( A 
C_  RR*  ->  ( x  e.  RR  ->  ( sup ( A ,  RR* ,  <  )  = +oo  ->  E. y  e.  A  x  <  y ) ) )
3736com23 81 . . 3  |-  ( A 
C_  RR*  ->  ( sup ( A ,  RR* ,  <  )  = +oo  ->  (
x  e.  RR  ->  E. y  e.  A  x  <  y ) ) )
3837ralrimdv 2806 . 2  |-  ( A 
C_  RR*  ->  ( sup ( A ,  RR* ,  <  )  = +oo  ->  A. x  e.  RR  E. y  e.  A  x  <  y
) )
3922, 38impbid 194 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 188    /\ wa 371    = wceq 1446    e. wcel 1889   A.wral 2739   E.wrex 2740    C_ wss 3406   class class class wbr 4405    Or wor 4757   supcsup 7959   RRcr 9543   +oocpnf 9677   RR*cxr 9679    < clt 9680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-po 4758  df-so 4759  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-sup 7961  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868
This theorem is referenced by:  supxrbnd2  11615  supxrbnd  11621  suplesup  37572  sge0pnffigt  38248
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