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Theorem ispointN 35550
 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 35549 . . 3 (𝐾𝐷𝑃 = {𝑥 ∣ ∃𝑎𝐴 𝑥 = {𝑎}})
43eleq2d 2834 . 2 (𝐾𝐷 → (𝑋𝑃𝑋 ∈ {𝑥 ∣ ∃𝑎𝐴 𝑥 = {𝑎}}))
5 snex 5035 . . . . 5 {𝑎} ∈ V
6 eleq1 2836 . . . . 5 (𝑋 = {𝑎} → (𝑋 ∈ V ↔ {𝑎} ∈ V))
75, 6mpbiri 248 . . . 4 (𝑋 = {𝑎} → 𝑋 ∈ V)
87rexlimivw 3175 . . 3 (∃𝑎𝐴 𝑋 = {𝑎} → 𝑋 ∈ V)
9 eqeq1 2773 . . . 4 (𝑥 = 𝑋 → (𝑥 = {𝑎} ↔ 𝑋 = {𝑎}))
109rexbidv 3198 . . 3 (𝑥 = 𝑋 → (∃𝑎𝐴 𝑥 = {𝑎} ↔ ∃𝑎𝐴 𝑋 = {𝑎}))
118, 10elab3 3506 . 2 (𝑋 ∈ {𝑥 ∣ ∃𝑎𝐴 𝑥 = {𝑎}} ↔ ∃𝑎𝐴 𝑋 = {𝑎})
124, 11syl6bb 276 1 (𝐾𝐷 → (𝑋𝑃 ↔ ∃𝑎𝐴 𝑋 = {𝑎}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   = wceq 1629   ∈ wcel 2143  {cab 2755  ∃wrex 3060  Vcvv 3348  {csn 4313  ‘cfv 6030  Atomscatm 35072  PointscpointsN 35303 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2145  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-rep 4901  ax-sep 4911  ax-nul 4919  ax-pr 5033  ax-un 7094 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1071  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-mo 2621  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ne 2942  df-ral 3064  df-rex 3065  df-reu 3066  df-rab 3068  df-v 3350  df-sbc 3585  df-csb 3680  df-dif 3723  df-un 3725  df-in 3727  df-ss 3734  df-nul 4061  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4572  df-iun 4653  df-br 4784  df-opab 4844  df-mpt 4861  df-id 5156  df-xp 5254  df-rel 5255  df-cnv 5256  df-co 5257  df-dm 5258  df-rn 5259  df-res 5260  df-ima 5261  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-pointsN 35310 This theorem is referenced by:  atpointN  35551  pointpsubN  35559
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