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Theorem infxrunb2 39584
Description: The infimum of an unbounded-below set of extended reals is minus infinity. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Assertion
Ref Expression
infxrunb2 |- ( A C_ RR* -> ( A. x e. RR E. y e. A y < x <-> inf ( A , RR* , < ) = -oo ) )
Distinct variable group:   y, A, x
Proof of Theorem infxrunb2
Dummy variables w z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1843 . . . . 5 |- F/ x A C_ RR*
2 nfra1 2941 . . . . 5 |- F/ x A. x e. RR E. y e. A y < x
31, 2nfan 1828 . . . 4 |- F/ x ( A C_ RR* /\ A. x e. RR E. y e. A y < x )
4 nfv 1843 . . . . 5 |- F/ y A C_ RR*
5 nfcv 2764 . . . . . 6 |- F/_ y RR
6 nfre1 3005 . . . . . 6 |- F/ y E. y e. A y < x
75, 6nfral 2945 . . . . 5 |- F/ y A. x e. RR E. y e. A y < x
84, 7nfan 1828 . . . 4 |- F/ y ( A C_ RR* /\ A. x e. RR E. y e. A y < x )
9 simpl 473 . . . 4 |- ( ( A C_ RR* /\ A. x e. RR E. y e. A y < x ) -> A C_ RR* )
10 mnfxr 10096 . . . . 5 |- -oo e. RR*
1110a1i 11 . . . 4 |- ( ( A C_ RR* /\ A. x e. RR E. y e. A y < x ) -> -oo e. RR* )
12 ssel2 3598 . . . . . . 7 |- ( ( A C_ RR* /\ x e. A ) -> x e. RR* )
13 nltmnf 11963 . . . . . . 7 |- ( x e. RR* -> -. x < -oo )
1412, 13syl 17 . . . . . 6 |- ( ( A C_ RR* /\ x e. A ) -> -. x < -oo )
1514ralrimiva 2966 . . . . 5 |- ( A C_ RR* -> A. x e. A -. x < -oo )
1615adantr 481 . . . 4 |- ( ( A C_ RR* /\ A. x e. RR E. y e. A y < x ) -> A. x e. A -. x < -oo )
17 ralimralim 39253 . . . . 5 |- ( A. x e. RR E. y e. A y < x -> A. x e. RR ( -oo < x -> E. y e. A y < x ) )
1817adantl 482 . . . 4 |- ( ( A C_ RR* /\ A. x e. RR E. y e. A y < x ) -> A. x e. RR ( -oo < x -> E. y e. A y < x ) )
193, 8, 9, 11, 16, 18infxr 39583 . . 3 |- ( ( A C_ RR* /\ A. x e. RR E. y e. A y < x ) -> inf ( A , RR* , < ) = -oo )
2019ex 450 . 2 |- ( A C_ RR* -> ( A. x e. RR E. y e. A y < x -> inf ( A , RR* , < ) = -oo ) )
21 rexr 10085 . . . . . 6 |- ( x e. RR -> x e. RR* )
2221adantl 482 . . . . 5 |- ( ( ( A C_ RR* /\ inf ( A , RR* , < ) = -oo ) /\ x e. RR ) -> x e. RR* )
23 simpl 473 . . . . . . 7 |- ( (inf ( A , RR* , < ) = -oo /\ x e. RR ) -> inf ( A , RR* , < ) = -oo )
24 mnflt 11957 . . . . . . . 8 |- ( x e. RR -> -oo < x )
2524adantl 482 . . . . . . 7 |- ( (inf ( A , RR* , < ) = -oo /\ x e. RR ) -> -oo < x )
2623, 25eqbrtrd 4675 . . . . . 6 |- ( (inf ( A , RR* , < ) = -oo /\ x e. RR ) -> inf ( A , RR* , < ) < x )
2726adantll 750 . . . . 5 |- ( ( ( A C_ RR* /\ inf ( A , RR* , < ) = -oo ) /\ x e. RR ) -> inf ( A , RR* , < ) < x )
28 xrltso 11974 . . . . . . 7 |- < Or RR*
2928a1i 11 . . . . . 6 |- ( ( ( A C_ RR* /\ inf ( A , RR* , < ) = -oo ) /\ x e. RR ) -> < Or RR* )
30 xrinfmss 12140 . . . . . . 7 |- ( A C_ RR* -> E. z e. RR* ( A. w e. A -. w < z /\ A. w e. RR* ( z < w -> E. y e. A y < w ) ) )
3130ad2antrr 762 . . . . . 6 |- ( ( ( A C_ RR* /\ inf ( A , RR* , < ) = -oo ) /\ x e. RR ) -> E. z e. RR* ( A. w e. A -. w < z /\ A. w e. RR* ( z < w -> E. y e. A y < w ) ) )
3229, 31infglb 8396 . . . . 5 |- ( ( ( A C_ RR* /\ inf ( A , RR* , < ) = -oo ) /\ x e. RR ) -> ( ( x e. RR* /\ inf ( A , RR* , < ) < x ) -> E. y e. A y < x ) )
3322, 27, 32mp2and 715 . . . 4 |- ( ( ( A C_ RR* /\ inf ( A , RR* , < ) = -oo ) /\ x e. RR ) -> E. y e. A y < x )
3433ralrimiva 2966 . . 3 |- ( ( A C_ RR* /\ inf ( A , RR* , < ) = -oo ) -> A. x e. RR E. y e. A y < x )
3534ex 450 . 2 |- ( A C_ RR* -> (inf ( A , RR* , < ) = -oo -> A. x e. RR E. y e. A y < x ) )
3620, 35impbid 202 1 |- ( A C_ RR* -> ( A. x e. RR E. y e. A y < x <-> inf ( A , RR* , < ) = -oo ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   class class class wbr 4653   Or wor 5034  infcinf 8347   RRcr 9935   -oocmnf 10072   RR*cxr 10073   < clt 10074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269
This theorem is referenced by:  infxrbnd2  39585  infleinf  39588  infxrunb3  39651  supminfxr  39694
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