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Theorem ispointN 37031
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 37030 . . 3 (𝐾𝐷𝑃 = {𝑥 ∣ ∃𝑎𝐴 𝑥 = {𝑎}})
43eleq2d 2878 . 2 (𝐾𝐷 → (𝑋𝑃𝑋 ∈ {𝑥 ∣ ∃𝑎𝐴 𝑥 = {𝑎}}))
5 snex 5300 . . . . 5 {𝑎} ∈ V
6 eleq1 2880 . . . . 5 (𝑋 = {𝑎} → (𝑋 ∈ V ↔ {𝑎} ∈ V))
75, 6mpbiri 261 . . . 4 (𝑋 = {𝑎} → 𝑋 ∈ V)
87rexlimivw 3244 . . 3 (∃𝑎𝐴 𝑋 = {𝑎} → 𝑋 ∈ V)
9 eqeq1 2805 . . . 4 (𝑥 = 𝑋 → (𝑥 = {𝑎} ↔ 𝑋 = {𝑎}))
109rexbidv 3259 . . 3 (𝑥 = 𝑋 → (∃𝑎𝐴 𝑥 = {𝑎} ↔ ∃𝑎𝐴 𝑋 = {𝑎}))
118, 10elab3 3625 . 2 (𝑋 ∈ {𝑥 ∣ ∃𝑎𝐴 𝑥 = {𝑎}} ↔ ∃𝑎𝐴 𝑋 = {𝑎})
124, 11syl6bb 290 1 (𝐾𝐷 → (𝑋𝑃 ↔ ∃𝑎𝐴 𝑋 = {𝑎}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2112  {cab 2779  wrex 3110  Vcvv 3444  {csn 4528  cfv 6328  Atomscatm 36552  PointscpointsN 36784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-pointsN 36791
This theorem is referenced by:  atpointN  37032  pointpsubN  37040
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