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Theorem uzinfmi 11161
 Description: Extract the lower bound of an upper set of integers as its infimum. Note that the " " argument turns supremum into infimum (for which we do not currently have a separate notation). (Contributed by NM, 7-Oct-2005.)
Hypothesis
Ref Expression
uzinfm.1
Assertion
Ref Expression
uzinfmi

Proof of Theorem uzinfmi
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uzssz 11101 . . . 4
2 zssre 10871 . . . 4
31, 2sstri 3513 . . 3
4 uzinfm.1 . . . . 5
5 uzid 11096 . . . . 5
64, 5ax-mp 5 . . . 4
7 eluzle 11094 . . . . 5
87rgen 2824 . . . 4
9 breq1 4450 . . . . . 6
109ralbidv 2903 . . . . 5
1110rspcev 3214 . . . 4
126, 8, 11mp2an 672 . . 3
13 lbinfm 10496 . . 3
143, 12, 13mp2an 672 . 2
15 lbreu 10493 . . . . 5
163, 12, 15mp2an 672 . . . 4
1710riota2 6268 . . . 4
186, 16, 17mp2an 672 . . 3
198, 18mpbi 208 . 2
2014, 19eqtri 2496 1
 Colors of variables: wff setvar class Syntax hints:   wb 184   wceq 1379   wcel 1767  wral 2814  wrex 2815  wreu 2816   wss 3476   class class class wbr 4447  ccnv 4998  cfv 5588  crio 6244  csup 7900  cr 9491   clt 9628   cle 9629  cz 10864  cuz 11082 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-pre-lttri 9566  ax-pre-lttrn 9567 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-sup 7901  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-neg 9808  df-z 10865  df-uz 11083 This theorem is referenced by:  nninfm  11162  nn0infm  11163
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