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Theorem ustdiag 18230
 Description: The diagonal set is included in any entourage, i.e. any point is -close to itself. Condition UI of [BourbakiTop1] p. II.1. (Contributed by Thierry Arnoux, 2-Dec-2017.)
Assertion
Ref Expression
ustdiag UnifOn

Proof of Theorem ustdiag
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 5750 . . . . . . 7 UnifOn
2 isust 18225 . . . . . . 7 UnifOn
31, 2syl 16 . . . . . 6 UnifOn UnifOn
43ibi 233 . . . . 5 UnifOn
54simp3d 971 . . . 4 UnifOn
6 sseq1 3361 . . . . . . . 8
76imbi1d 309 . . . . . . 7
87ralbidv 2717 . . . . . 6
9 ineq1 3527 . . . . . . . 8
109eleq1d 2501 . . . . . . 7
1110ralbidv 2717 . . . . . 6
12 sseq2 3362 . . . . . . 7
13 cnveq 5038 . . . . . . . 8
1413eleq1d 2501 . . . . . . 7
15 sseq2 3362 . . . . . . . 8
1615rexbidv 2718 . . . . . . 7
1712, 14, 163anbi123d 1254 . . . . . 6
188, 11, 173anbi123d 1254 . . . . 5
1918rspcv 3040 . . . 4
205, 19mpan9 456 . . 3 UnifOn
2120simp3d 971 . 2 UnifOn
2221simp1d 969 1 UnifOn
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2697  wrex 2698  cvv 2948   cin 3311   wss 3312  cpw 3791   cid 4485   cxp 4868  ccnv 4869   cres 4872   ccom 4874  cfv 5446  UnifOncust 18221 This theorem is referenced by:  ustssco  18236  ustref  18240  ustelimasn  18244  trust  18251  ustuqtop3  18265 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-res 4882  df-iota 5410  df-fun 5448  df-fv 5454  df-ust 18222
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