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Theorem atpointN 37380
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 4526 . . . . 5 (𝑥 = 𝑋 → {𝑥} = {𝑋})
32rspceeqv 3541 . . . 4 ((𝑋𝐴 ∧ {𝑋} = {𝑋}) → ∃𝑥𝐴 {𝑋} = {𝑥})
41, 3mpan2 691 . . 3 (𝑋𝐴 → ∃𝑥𝐴 {𝑋} = {𝑥})
54adantl 485 . 2 ((𝐾𝐷𝑋𝐴) → ∃𝑥𝐴 {𝑋} = {𝑥})
6 ispoint.a . . . 4 𝐴 = (Atoms‘𝐾)
7 ispoint.p . . . 4 𝑃 = (Points‘𝐾)
86, 7ispointN 37379 . . 3 (𝐾𝐷 → ({𝑋} ∈ 𝑃 ↔ ∃𝑥𝐴 {𝑋} = {𝑥}))
98adantr 484 . 2 ((𝐾𝐷𝑋𝐴) → ({𝑋} ∈ 𝑃 ↔ ∃𝑥𝐴 {𝑋} = {𝑥}))
105, 9mpbird 260 1 ((𝐾𝐷𝑋𝐴) → {𝑋} ∈ 𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114  wrex 3054  {csn 4516  cfv 6339  Atomscatm 36900  PointscpointsN 37132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-pointsN 37139
This theorem is referenced by: (None)
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