Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  uzinfmiOLD Structured version   Visualization version   Unicode version

Theorem uzinfmiOLD 11239
 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.) Obsolete version of uzinfi 11238 as of 4-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
uzinfi.1
Assertion
Ref Expression
uzinfmiOLD

Proof of Theorem uzinfmiOLD
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uzssz 11178 . . . 4
2 zssre 10944 . . . 4
31, 2sstri 3441 . . 3
4 uzinfi.1 . . . . 5
5 uzid 11173 . . . . 5
64, 5ax-mp 5 . . . 4
7 eluzle 11171 . . . . 5
87rgen 2747 . . . 4
9 breq1 4405 . . . . . 6
109ralbidv 2827 . . . . 5
1110rspcev 3150 . . . 4
126, 8, 11mp2an 678 . . 3
13 lbinfmOLD 10560 . . 3
143, 12, 13mp2an 678 . 2
15 lbreu 10556 . . . . 5
163, 12, 15mp2an 678 . . . 4
1710riota2 6274 . . . 4
186, 16, 17mp2an 678 . . 3
198, 18mpbi 212 . 2
2014, 19eqtri 2473 1
 Colors of variables: wff setvar class Syntax hints:   wb 188   wceq 1444   wcel 1887  wral 2737  wrex 2738  wreu 2739   wss 3404   class class class wbr 4402  ccnv 4833  cfv 5582  crio 6251  csup 7954  cr 9538   clt 9675   cle 9676  cz 10937  cuz 11159 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-pre-lttri 9613  ax-pre-lttrn 9614 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-po 4755  df-so 4756  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-sup 7956  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-neg 9863  df-z 10938  df-uz 11160 This theorem is referenced by:  nninfmOLD  11242  nn0infmOLD  11243
 Copyright terms: Public domain W3C validator