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Theorem atpointN 34047
 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 2610 . . . 4 {𝑋} = {𝑋}
2 sneq 4135 . . . . . 6 (𝑥 = 𝑋 → {𝑥} = {𝑋})
32eqeq2d 2620 . . . . 5 (𝑥 = 𝑋 → ({𝑋} = {𝑥} ↔ {𝑋} = {𝑋}))
43rspcev 3282 . . . 4 ((𝑋𝐴 ∧ {𝑋} = {𝑋}) → ∃𝑥𝐴 {𝑋} = {𝑥})
51, 4mpan2 703 . . 3 (𝑋𝐴 → ∃𝑥𝐴 {𝑋} = {𝑥})
65adantl 481 . 2 ((𝐾𝐷𝑋𝐴) → ∃𝑥𝐴 {𝑋} = {𝑥})
7 ispoint.a . . . 4 𝐴 = (Atoms‘𝐾)
8 ispoint.p . . . 4 𝑃 = (Points‘𝐾)
97, 8ispointN 34046 . . 3 (𝐾𝐷 → ({𝑋} ∈ 𝑃 ↔ ∃𝑥𝐴 {𝑋} = {𝑥}))
109adantr 480 . 2 ((𝐾𝐷𝑋𝐴) → ({𝑋} ∈ 𝑃 ↔ ∃𝑥𝐴 {𝑋} = {𝑥}))
116, 10mpbird 246 1 ((𝐾𝐷𝑋𝐴) → {𝑋} ∈ 𝑃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  {csn 4125  ‘cfv 5804  Atomscatm 33568  PointscpointsN 33799 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-pointsN 33806 This theorem is referenced by: (None)
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