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Theorem ispointN 39744
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 39743 . . 3 (𝐾𝐷𝑃 = {𝑥 ∣ ∃𝑎𝐴 𝑥 = {𝑎}})
43eleq2d 2827 . 2 (𝐾𝐷 → (𝑋𝑃𝑋 ∈ {𝑥 ∣ ∃𝑎𝐴 𝑥 = {𝑎}}))
5 vsnex 5434 . . . . 5 {𝑎} ∈ V
6 eleq1 2829 . . . . 5 (𝑋 = {𝑎} → (𝑋 ∈ V ↔ {𝑎} ∈ V))
75, 6mpbiri 258 . . . 4 (𝑋 = {𝑎} → 𝑋 ∈ V)
87rexlimivw 3151 . . 3 (∃𝑎𝐴 𝑋 = {𝑎} → 𝑋 ∈ V)
9 eqeq1 2741 . . . 4 (𝑥 = 𝑋 → (𝑥 = {𝑎} ↔ 𝑋 = {𝑎}))
109rexbidv 3179 . . 3 (𝑥 = 𝑋 → (∃𝑎𝐴 𝑥 = {𝑎} ↔ ∃𝑎𝐴 𝑋 = {𝑎}))
118, 10elab3 3686 . 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 1540  wcel 2108  {cab 2714  wrex 3070  Vcvv 3480  {csn 4626  cfv 6561  Atomscatm 39264  PointscpointsN 39497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-pointsN 39504
This theorem is referenced by:  atpointN  39745  pointpsubN  39753
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