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Theorem elee 25825
 Description: Membership in a Euclidean space. We define Euclidean space here using Cartesian coordinates over space. We later abstract away from this using Tarski's geometry axioms, so this exact definition is unimportant. (Contributed by Scott Fenton, 3-Jun-2013.)
Assertion
Ref Expression
elee

Proof of Theorem elee
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 oveq2 6081 . . . . 5
21oveq2d 6089 . . . 4
3 df-ee 25822 . . . 4
4 ovex 6098 . . . 4
52, 3, 4fvmpt 5798 . . 3
65eleq2d 2502 . 2
7 reex 9073 . . 3
8 ovex 6098 . . 3
97, 8elmap 7034 . 2
106, 9syl6bb 253 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wceq 1652   wcel 1725  wf 5442  cfv 5446  (class class class)co 6073   cmap 7010  cr 8981  c1 8983  cn 9992  cfz 11035  cee 25819 This theorem is referenced by:  mptelee  25826  eleei  25828  axlowdimlem5  25877  axlowdimlem7  25879  axlowdimlem10  25882  axlowdimlem14  25886  axlowdim1  25890 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  ax-cnex 9038  ax-resscn 9039 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-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-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-ee 25822
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