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Theorem ispointN 39125
Description: The predicate "is a point". (Contributed by NM, 2-Oct-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
ispoint.a 𝐴 = (Atomsβ€˜πΎ)
ispoint.p 𝑃 = (Pointsβ€˜πΎ)
Assertion
Ref Expression
ispointN (𝐾 ∈ 𝐷 β†’ (𝑋 ∈ 𝑃 ↔ βˆƒπ‘Ž ∈ 𝐴 𝑋 = {π‘Ž}))
Distinct variable groups:   𝐴,π‘Ž   𝑋,π‘Ž
Allowed substitution hints:   𝐷(π‘Ž)   𝑃(π‘Ž)   𝐾(π‘Ž)

Proof of Theorem ispointN
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 ispoint.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
2 ispoint.p . . . 4 𝑃 = (Pointsβ€˜πΎ)
31, 2pointsetN 39124 . . 3 (𝐾 ∈ 𝐷 β†’ 𝑃 = {π‘₯ ∣ βˆƒπ‘Ž ∈ 𝐴 π‘₯ = {π‘Ž}})
43eleq2d 2813 . 2 (𝐾 ∈ 𝐷 β†’ (𝑋 ∈ 𝑃 ↔ 𝑋 ∈ {π‘₯ ∣ βˆƒπ‘Ž ∈ 𝐴 π‘₯ = {π‘Ž}}))
5 vsnex 5422 . . . . 5 {π‘Ž} ∈ V
6 eleq1 2815 . . . . 5 (𝑋 = {π‘Ž} β†’ (𝑋 ∈ V ↔ {π‘Ž} ∈ V))
75, 6mpbiri 258 . . . 4 (𝑋 = {π‘Ž} β†’ 𝑋 ∈ V)
87rexlimivw 3145 . . 3 (βˆƒπ‘Ž ∈ 𝐴 𝑋 = {π‘Ž} β†’ 𝑋 ∈ V)
9 eqeq1 2730 . . . 4 (π‘₯ = 𝑋 β†’ (π‘₯ = {π‘Ž} ↔ 𝑋 = {π‘Ž}))
109rexbidv 3172 . . 3 (π‘₯ = 𝑋 β†’ (βˆƒπ‘Ž ∈ 𝐴 π‘₯ = {π‘Ž} ↔ βˆƒπ‘Ž ∈ 𝐴 𝑋 = {π‘Ž}))
118, 10elab3 3671 . 2 (𝑋 ∈ {π‘₯ ∣ βˆƒπ‘Ž ∈ 𝐴 π‘₯ = {π‘Ž}} ↔ βˆƒπ‘Ž ∈ 𝐴 𝑋 = {π‘Ž})
124, 11bitrdi 287 1 (𝐾 ∈ 𝐷 β†’ (𝑋 ∈ 𝑃 ↔ βˆƒπ‘Ž ∈ 𝐴 𝑋 = {π‘Ž}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  {cab 2703  βˆƒwrex 3064  Vcvv 3468  {csn 4623  β€˜cfv 6536  Atomscatm 38645  PointscpointsN 38878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-pointsN 38885
This theorem is referenced by:  atpointN  39126  pointpsubN  39134
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