Theorem atpointN 40003

Description: The singleton of an atom is a point. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ispoint.a	⊢ 𝐴 = (Atoms‘𝐾)
ispoint.p	⊢ 𝑃 = (Points‘𝐾)

Assertion

Ref	Expression
atpointN	⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐴) → {𝑋} ∈ 𝑃)

Proof of Theorem atpointN

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2736	. . . 4 ⊢ {𝑋} = {𝑋}
2		sneq 4590	. . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
3	2	rspceeqv 3599	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ {𝑋} = {𝑋}) → ∃𝑥 ∈ 𝐴 {𝑋} = {𝑥})
4	1, 3	mpan2 691	. . 3 ⊢ (𝑋 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 {𝑋} = {𝑥})
5	4	adantl 481	. 2 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 {𝑋} = {𝑥})
6		ispoint.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
7		ispoint.p	. . . 4 ⊢ 𝑃 = (Points‘𝐾)
8	6, 7	ispointN 40002	. . 3 ⊢ (𝐾 ∈ 𝐷 → ({𝑋} ∈ 𝑃 ↔ ∃𝑥 ∈ 𝐴 {𝑋} = {𝑥}))
9	8	adantr 480	. 2 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐴) → ({𝑋} ∈ 𝑃 ↔ ∃𝑥 ∈ 𝐴 {𝑋} = {𝑥}))
10	5, 9	mpbird 257	1 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐴) → {𝑋} ∈ 𝑃)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3060  {csn 4580  ‘cfv 6492  Atomscatm 39523  PointscpointsN 39755

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 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-pointsN 39762

This theorem is referenced by: (None)