Theorem ispointN 39743

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 39742	. . 3 ⊢ (𝐾 ∈ 𝐷 → 𝑃 = {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = {𝑎}})
4	3	eleq2d 2815	. 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑃 ↔ 𝑋 ∈ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = {𝑎}}))
5		vsnex 5392	. . . . 5 ⊢ {𝑎} ∈ V
6		eleq1 2817	. . . . 5 ⊢ (𝑋 = {𝑎} → (𝑋 ∈ V ↔ {𝑎} ∈ V))
7	5, 6	mpbiri 258	. . . 4 ⊢ (𝑋 = {𝑎} → 𝑋 ∈ V)
8	7	rexlimivw 3131	. . 3 ⊢ (∃𝑎 ∈ 𝐴 𝑋 = {𝑎} → 𝑋 ∈ V)
9		eqeq1 2734	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 = {𝑎} ↔ 𝑋 = {𝑎}))
10	9	rexbidv 3158	. . 3 ⊢ (𝑥 = 𝑋 → (∃𝑎 ∈ 𝐴 𝑥 = {𝑎} ↔ ∃𝑎 ∈ 𝐴 𝑋 = {𝑎}))
11	8, 10	elab3 3656	. 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 1540   ∈ wcel 2109  {cab 2708  ∃wrex 3054  Vcvv 3450  {csn 4592  ‘cfv 6514  Atomscatm 39263  PointscpointsN 39496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-pointsN 39503

This theorem is referenced by:  atpointN  39744  pointpsubN  39752