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Theorem atpointN 37757
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 2738 . . . 4 {𝑋} = {𝑋}
2 sneq 4571 . . . . 5 (𝑥 = 𝑋 → {𝑥} = {𝑋})
32rspceeqv 3575 . . . 4 ((𝑋𝐴 ∧ {𝑋} = {𝑋}) → ∃𝑥𝐴 {𝑋} = {𝑥})
41, 3mpan2 688 . . 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 37756 . . 3 (𝐾𝐷 → ({𝑋} ∈ 𝑃 ↔ ∃𝑥𝐴 {𝑋} = {𝑥}))
98adantr 481 . 2 ((𝐾𝐷𝑋𝐴) → ({𝑋} ∈ 𝑃 ↔ ∃𝑥𝐴 {𝑋} = {𝑥}))
105, 9mpbird 256 1 ((𝐾𝐷𝑋𝐴) → {𝑋} ∈ 𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  wrex 3065  {csn 4561  cfv 6433  Atomscatm 37277  PointscpointsN 37509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-pointsN 37516
This theorem is referenced by: (None)
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