Theorem atpointN 40616

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 2760	. . . 4 ⊢ {𝑋} = {𝑋}
2		sneq 4594	. . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
3	2	rspceeqv 3599	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ {𝑋} = {𝑋}) → ∃𝑥 ∈ 𝐴 {𝑋} = {𝑥})
4	1, 3	mpan2 704	. . 3 ⊢ (𝑋 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 {𝑋} = {𝑥})
5	4	adantl 487	. 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 40615	. . 3 ⊢ (𝐾 ∈ 𝐷 → ({𝑋} ∈ 𝑃 ↔ ∃𝑥 ∈ 𝐴 {𝑋} = {𝑥}))
9	8	adantr 486	. 2 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐴) → ({𝑋} ∈ 𝑃 ↔ ∃𝑥 ∈ 𝐴 {𝑋} = {𝑥}))
10	5, 9	mpbird 260	1 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐴) → {𝑋} ∈ 𝑃)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3086  {csn 4584  ‘cfv 6533  Atomscatm 40136  PointscpointsN 40368

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fv 6541  df-pointsN 40375

This theorem is used by: (None)