Theorem islp2 23101

Description: The predicate "𝑃 is a limit point of 𝑆 " in terms of neighborhoods. Definition of limit point in [Munkres] p. 97. Although Munkres uses open neighborhoods, it also works for our more general neighborhoods. (Contributed by NM, 26-Feb-2007.) (Proof shortened by Mario Carneiro, 25-Dec-2016.)

Hypothesis

Ref	Expression
lpfval.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
islp2	⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅))

Distinct variable groups:   𝑛,𝐽   𝑃,𝑛   𝑆,𝑛   𝑛,𝑋

Proof of Theorem islp2

Step	Hyp	Ref	Expression
1		lpfval.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
2	1	islp 23096	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃}))))
3	2	3adant3 1133	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃}))))
4		ssdifss 4094	. . 3 ⊢ (𝑆 ⊆ 𝑋 → (𝑆 ∖ {𝑃}) ⊆ 𝑋)
5	1	neindisj2 23079	. . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ∖ {𝑃}) ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃})) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅))
6	4, 5	syl3an2 1165	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃})) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅))
7	3, 6	bitrd 279	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052   ∖ cdif 3900   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287  {csn 4582  ∪ cuni 4865  ‘cfv 6500  Topctop 22849  clsccl 22974  neicnei 23053  limPtclp 23090

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-pow 5312  ax-pr 5379  ax-un 7690

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-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  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-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-top 22850  df-cld 22975  df-ntr 22976  df-cls 22977  df-nei 23054  df-lp 23092

This theorem is referenced by:  clslp  23104  lpbl  24459  reperflem  24775  islptre  45979  islpcn  45997