Theorem atpointN 40119

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 2737	. . . 4 ⊢ {𝑋} = {𝑋}
2		sneq 4592	. . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
3	2	rspceeqv 3601	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ {𝑋} = {𝑋}) → ∃𝑥 ∈ 𝐴 {𝑋} = {𝑥})
4	1, 3	mpan2 692	. . 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 40118	. . 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 1542   ∈ wcel 2114  ∃wrex 3062  {csn 4582  ‘cfv 6500  Atomscatm 39639  PointscpointsN 39871

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-pointsN 39878

This theorem is referenced by: (None)