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Theorem ispointN 36758
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 36757 . . 3 (𝐾𝐷𝑃 = {𝑥 ∣ ∃𝑎𝐴 𝑥 = {𝑎}})
43eleq2d 2895 . 2 (𝐾𝐷 → (𝑋𝑃𝑋 ∈ {𝑥 ∣ ∃𝑎𝐴 𝑥 = {𝑎}}))
5 snex 5322 . . . . 5 {𝑎} ∈ V
6 eleq1 2897 . . . . 5 (𝑋 = {𝑎} → (𝑋 ∈ V ↔ {𝑎} ∈ V))
75, 6mpbiri 259 . . . 4 (𝑋 = {𝑎} → 𝑋 ∈ V)
87rexlimivw 3279 . . 3 (∃𝑎𝐴 𝑋 = {𝑎} → 𝑋 ∈ V)
9 eqeq1 2822 . . . 4 (𝑥 = 𝑋 → (𝑥 = {𝑎} ↔ 𝑋 = {𝑎}))
109rexbidv 3294 . . 3 (𝑥 = 𝑋 → (∃𝑎𝐴 𝑥 = {𝑎} ↔ ∃𝑎𝐴 𝑋 = {𝑎}))
118, 10elab3 3671 . 2 (𝑋 ∈ {𝑥 ∣ ∃𝑎𝐴 𝑥 = {𝑎}} ↔ ∃𝑎𝐴 𝑋 = {𝑎})
124, 11syl6bb 288 1 (𝐾𝐷 → (𝑋𝑃 ↔ ∃𝑎𝐴 𝑋 = {𝑎}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  wcel 2105  {cab 2796  wrex 3136  Vcvv 3492  {csn 4557  cfv 6348  Atomscatm 36279  PointscpointsN 36511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-pointsN 36518
This theorem is referenced by:  atpointN  36759  pointpsubN  36767
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