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Theorem ispointN 39724
Description: The predicate "is a point". (Contributed by NM, 2-Oct-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
ispoint.a 𝐴 = (Atoms‘𝐾)
ispoint.p 𝑃 = (Points‘𝐾)
Assertion
Ref Expression
ispointN (𝐾𝐷 → (𝑋𝑃 ↔ ∃𝑎𝐴 𝑋 = {𝑎}))
Distinct variable groups:   𝐴,𝑎   𝑋,𝑎
Allowed substitution hints:   𝐷(𝑎)   𝑃(𝑎)   𝐾(𝑎)

Proof of Theorem ispointN
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ispoint.a . . . 4 𝐴 = (Atoms‘𝐾)
2 ispoint.p . . . 4 𝑃 = (Points‘𝐾)
31, 2pointsetN 39723 . . 3 (𝐾𝐷𝑃 = {𝑥 ∣ ∃𝑎𝐴 𝑥 = {𝑎}})
43eleq2d 2824 . 2 (𝐾𝐷 → (𝑋𝑃𝑋 ∈ {𝑥 ∣ ∃𝑎𝐴 𝑥 = {𝑎}}))
5 vsnex 5439 . . . . 5 {𝑎} ∈ V
6 eleq1 2826 . . . . 5 (𝑋 = {𝑎} → (𝑋 ∈ V ↔ {𝑎} ∈ V))
75, 6mpbiri 258 . . . 4 (𝑋 = {𝑎} → 𝑋 ∈ V)
87rexlimivw 3148 . . 3 (∃𝑎𝐴 𝑋 = {𝑎} → 𝑋 ∈ V)
9 eqeq1 2738 . . . 4 (𝑥 = 𝑋 → (𝑥 = {𝑎} ↔ 𝑋 = {𝑎}))
109rexbidv 3176 . . 3 (𝑥 = 𝑋 → (∃𝑎𝐴 𝑥 = {𝑎} ↔ ∃𝑎𝐴 𝑋 = {𝑎}))
118, 10elab3 3688 . 2 (𝑋 ∈ {𝑥 ∣ ∃𝑎𝐴 𝑥 = {𝑎}} ↔ ∃𝑎𝐴 𝑋 = {𝑎})
124, 11bitrdi 287 1 (𝐾𝐷 → (𝑋𝑃 ↔ ∃𝑎𝐴 𝑋 = {𝑎}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1536  wcel 2105  {cab 2711  wrex 3067  Vcvv 3477  {csn 4630  cfv 6562  Atomscatm 39244  PointscpointsN 39477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-pointsN 39484
This theorem is referenced by:  atpointN  39725  pointpsubN  39733
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