Theorem ispointN 39914

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	⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑃 ↔ ∃𝑎 ∈ 𝐴 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.

Step	Hyp	Ref	Expression
1		ispoint.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
2		ispoint.p	. . . 4 ⊢ 𝑃 = (Points‘𝐾)
3	1, 2	pointsetN 39913	. . 3 ⊢ (𝐾 ∈ 𝐷 → 𝑃 = {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = {𝑎}})
4	3	eleq2d 2819	. 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑃 ↔ 𝑋 ∈ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = {𝑎}}))
5		vsnex 5376	. . . . 5 ⊢ {𝑎} ∈ V
6		eleq1 2821	. . . . 5 ⊢ (𝑋 = {𝑎} → (𝑋 ∈ V ↔ {𝑎} ∈ V))
7	5, 6	mpbiri 258	. . . 4 ⊢ (𝑋 = {𝑎} → 𝑋 ∈ V)
8	7	rexlimivw 3130	. . 3 ⊢ (∃𝑎 ∈ 𝐴 𝑋 = {𝑎} → 𝑋 ∈ V)
9		eqeq1 2737	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 = {𝑎} ↔ 𝑋 = {𝑎}))
10	9	rexbidv 3157	. . 3 ⊢ (𝑥 = 𝑋 → (∃𝑎 ∈ 𝐴 𝑥 = {𝑎} ↔ ∃𝑎 ∈ 𝐴 𝑋 = {𝑎}))
11	8, 10	elab3 3638	. 2 ⊢ (𝑋 ∈ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = {𝑎}} ↔ ∃𝑎 ∈ 𝐴 𝑋 = {𝑎})
12	4, 11	bitrdi 287	1 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑃 ↔ ∃𝑎 ∈ 𝐴 𝑋 = {𝑎}))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2113  {cab 2711  ∃wrex 3057  Vcvv 3437  {csn 4577  ‘cfv 6489  Atomscatm 39435  PointscpointsN 39667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6445  df-fun 6491  df-fv 6497  df-pointsN 39674

This theorem is referenced by:  atpointN  39915  pointpsubN  39923