Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  atpointN Structured version   Visualization version   GIF version

Theorem atpointN 36759
Description: The singleton of an atom is a point. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
ispoint.a 𝐴 = (Atoms‘𝐾)
ispoint.p 𝑃 = (Points‘𝐾)
Assertion
Ref Expression
atpointN ((𝐾𝐷𝑋𝐴) → {𝑋} ∈ 𝑃)

Proof of Theorem atpointN
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2818 . . . 4 {𝑋} = {𝑋}
2 sneq 4567 . . . . 5 (𝑥 = 𝑋 → {𝑥} = {𝑋})
32rspceeqv 3635 . . . 4 ((𝑋𝐴 ∧ {𝑋} = {𝑋}) → ∃𝑥𝐴 {𝑋} = {𝑥})
41, 3mpan2 687 . . 3 (𝑋𝐴 → ∃𝑥𝐴 {𝑋} = {𝑥})
54adantl 482 . 2 ((𝐾𝐷𝑋𝐴) → ∃𝑥𝐴 {𝑋} = {𝑥})
6 ispoint.a . . . 4 𝐴 = (Atoms‘𝐾)
7 ispoint.p . . . 4 𝑃 = (Points‘𝐾)
86, 7ispointN 36758 . . 3 (𝐾𝐷 → ({𝑋} ∈ 𝑃 ↔ ∃𝑥𝐴 {𝑋} = {𝑥}))
98adantr 481 . 2 ((𝐾𝐷𝑋𝐴) → ({𝑋} ∈ 𝑃 ↔ ∃𝑥𝐴 {𝑋} = {𝑥}))
105, 9mpbird 258 1 ((𝐾𝐷𝑋𝐴) → {𝑋} ∈ 𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wrex 3136  {csn 4557  cfv 6348  Atomscatm 36279  PointscpointsN 36511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-pointsN 36518
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator