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Theorem supxrunb2 10686
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 3208 . . . . . . . 8  |-  ( A 
C_  RR*  ->  ( z  e.  A  ->  z  e. 
RR* ) )
2 pnfnlt 10514 . . . . . . . 8  |-  ( z  e.  RR*  ->  -.  +oo  <  z )
31, 2syl6 29 . . . . . . 7  |-  ( A 
C_  RR*  ->  ( z  e.  A  ->  -.  +oo  <  z ) )
43ralrimiv 2659 . . . . . 6  |-  ( A 
C_  RR*  ->  A. z  e.  A  -.  +oo  <  z )
54adantr 451 . . . . 5  |-  ( ( A  C_  RR*  /\  A. x  e.  RR  E. y  e.  A  x  <  y )  ->  A. z  e.  A  -.  +oo  <  z )
6 breq1 4063 . . . . . . . . . . . . . . 15  |-  ( x  =  z  ->  (
x  <  y  <->  z  <  y ) )
76rexbidv 2598 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  ( E. y  e.  A  x  <  y  <->  E. y  e.  A  z  <  y ) )
87rspcva 2916 . . . . . . . . . . . . 13  |-  ( ( z  e.  RR  /\  A. x  e.  RR  E. y  e.  A  x  <  y )  ->  E. y  e.  A  z  <  y )
98adantrr 697 . . . . . . . . . . . 12  |-  ( ( z  e.  RR  /\  ( A. x  e.  RR  E. y  e.  A  x  <  y  /\  A  C_ 
RR* ) )  ->  E. y  e.  A  z  <  y )
109ancoms 439 . . . . . . . . . . 11  |-  ( ( ( A. x  e.  RR  E. y  e.  A  x  <  y  /\  A  C_  RR* )  /\  z  e.  RR )  ->  E. y  e.  A  z  <  y )
1110exp31 587 . . . . . . . . . 10  |-  ( A. x  e.  RR  E. y  e.  A  x  <  y  ->  ( A  C_  RR* 
->  ( z  e.  RR  ->  E. y  e.  A  z  <  y ) ) )
1211a1dd 42 . . . . . . . . 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 80 . . . . . . . 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 74 . . . . . . 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 418 . . . . . 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 2659 . . . . 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 518 . . . 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 10502 . . . . 5  |-  +oo  e.  RR*
19 supxr 10678 . . . . 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 662 . . . 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 456 . . 3  |-  ( ( A  C_  RR*  /\  A. x  e.  RR  E. y  e.  A  x  <  y )  ->  sup ( A ,  RR* ,  <  )  =  +oo )
2221ex 423 . 2  |-  ( A 
C_  RR*  ->  ( A. x  e.  RR  E. y  e.  A  x  <  y  ->  sup ( A ,  RR* ,  <  )  = 
+oo ) )
23 rexr 8922 . . . . . . 7  |-  ( x  e.  RR  ->  x  e.  RR* )
2423ad2antlr 707 . . . . . 6  |-  ( ( ( A  C_  RR*  /\  x  e.  RR )  /\  sup ( A ,  RR* ,  <  )  =  +oo )  ->  x  e.  RR* )
25 ltpnf 10510 . . . . . . . . 9  |-  ( x  e.  RR  ->  x  <  +oo )
26 breq2 4064 . . . . . . . . 9  |-  ( sup ( A ,  RR* ,  <  )  =  +oo  ->  ( x  <  sup ( A ,  RR* ,  <  )  <-> 
x  <  +oo ) )
2725, 26syl5ibr 212 . . . . . . . 8  |-  ( sup ( A ,  RR* ,  <  )  =  +oo  ->  ( x  e.  RR  ->  x  <  sup ( A ,  RR* ,  <  ) ) )
2827impcom 419 . . . . . . 7  |-  ( ( x  e.  RR  /\  sup ( A ,  RR* ,  <  )  =  +oo )  ->  x  <  sup ( A ,  RR* ,  <  ) )
2928adantll 694 . . . . . 6  |-  ( ( ( A  C_  RR*  /\  x  e.  RR )  /\  sup ( A ,  RR* ,  <  )  =  +oo )  ->  x  <  sup ( A ,  RR* ,  <  ) )
30 xrltso 10522 . . . . . . . 8  |-  <  Or  RR*
3130a1i 10 . . . . . . 7  |-  ( ( ( A  C_  RR*  /\  x  e.  RR )  /\  sup ( A ,  RR* ,  <  )  =  +oo )  ->  <  Or  RR* )
32 xrsupss 10674 . . . . . . . 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 706 . . . . . . 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 7256 . . . . . 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 660 . . . . 5  |-  ( ( ( A  C_  RR*  /\  x  e.  RR )  /\  sup ( A ,  RR* ,  <  )  =  +oo )  ->  E. y  e.  A  x  <  y )
3635exp31 587 . . . 4  |-  ( A 
C_  RR*  ->  ( x  e.  RR  ->  ( sup ( A ,  RR* ,  <  )  =  +oo  ->  E. y  e.  A  x  <  y ) ) )
3736com23 72 . . 3  |-  ( A 
C_  RR*  ->  ( sup ( A ,  RR* ,  <  )  =  +oo  ->  (
x  e.  RR  ->  E. y  e.  A  x  <  y ) ) )
3837ralrimdv 2666 . 2  |-  ( A 
C_  RR*  ->  ( sup ( A ,  RR* ,  <  )  =  +oo  ->  A. x  e.  RR  E. y  e.  A  x  <  y
) )
3922, 38impbid 183 1  |-  ( A 
C_  RR*  ->  ( A. x  e.  RR  E. y  e.  A  x  <  y  <->  sup ( A ,  RR* ,  <  )  =  +oo ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1633    e. wcel 1701   A.wral 2577   E.wrex 2578    C_ wss 3186   class class class wbr 4060    Or wor 4350   supcsup 7238   RRcr 8781    +oocpnf 8909   RR*cxr 8911    < clt 8912
This theorem is referenced by:  supxrbnd2  10688  supxrbnd  10694
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-po 4351  df-so 4352  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-riota 6346  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-sup 7239  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085
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