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Theorem elcls 22447
Description: Membership in a closure. Theorem 6.5(a) of [Munkres] p. 95. (Contributed by NM, 22-Feb-2007.)
Hypothesis
Ref Expression
clscld.1 𝑋 = 𝐽
Assertion
Ref Expression
elcls ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑥𝐽 (𝑃𝑥 → (𝑥𝑆) ≠ ∅)))
Distinct variable groups:   𝑥,𝐽   𝑥,𝑃   𝑥,𝑆   𝑥,𝑋

Proof of Theorem elcls
StepHypRef Expression
1 clscld.1 . . . . . . . 8 𝑋 = 𝐽
21cmclsopn 22436 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
323adant3 1133 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
43adantr 482 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
5 eldif 3924 . . . . . . 7 (𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ↔ (𝑃𝑋 ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
65biimpri 227 . . . . . 6 ((𝑃𝑋 ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
763ad2antl3 1188 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → 𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
8 simpr 486 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆𝑋)
91sscls 22430 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
108, 9ssind 4196 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ (𝑋 ∩ ((cls‘𝐽)‘𝑆)))
11 dfin4 4231 . . . . . . . . . 10 (𝑋 ∩ ((cls‘𝐽)‘𝑆)) = (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
1210, 11sseqtrdi 3998 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆))))
13 reldisj 4415 . . . . . . . . . 10 (𝑆𝑋 → ((𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅ ↔ 𝑆 ⊆ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))))
1413adantl 483 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅ ↔ 𝑆 ⊆ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))))
1512, 14mpbird 257 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅)
16 nne 2944 . . . . . . . . 9 (¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅ ↔ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) = ∅)
17 incom 4165 . . . . . . . . . 10 ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) = (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆)))
1817eqeq1i 2738 . . . . . . . . 9 (((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) = ∅ ↔ (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅)
1916, 18bitri 275 . . . . . . . 8 (¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅ ↔ (𝑆 ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑆))) = ∅)
2015, 19sylibr 233 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)
21203adant3 1133 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)
2221adantr 482 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)
23 eleq2 2823 . . . . . . 7 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → (𝑃𝑥𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆))))
24 ineq1 4169 . . . . . . . . 9 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → (𝑥𝑆) = ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆))
2524neeq1d 3000 . . . . . . . 8 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → ((𝑥𝑆) ≠ ∅ ↔ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅))
2625notbid 318 . . . . . . 7 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → (¬ (𝑥𝑆) ≠ ∅ ↔ ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅))
2723, 26anbi12d 632 . . . . . 6 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑆)) → ((𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅) ↔ (𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∧ ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)))
2827rspcev 3583 . . . . 5 (((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽 ∧ (𝑃 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∧ ¬ ((𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∩ 𝑆) ≠ ∅)) → ∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅))
294, 7, 22, 28syl12anc 836 . . . 4 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) ∧ ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) → ∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅))
30 incom 4165 . . . . . . . . . . . . 13 (𝑆𝑥) = (𝑥𝑆)
3130eqeq1i 2738 . . . . . . . . . . . 12 ((𝑆𝑥) = ∅ ↔ (𝑥𝑆) = ∅)
32 df-ne 2941 . . . . . . . . . . . . 13 ((𝑥𝑆) ≠ ∅ ↔ ¬ (𝑥𝑆) = ∅)
3332con2bii 358 . . . . . . . . . . . 12 ((𝑥𝑆) = ∅ ↔ ¬ (𝑥𝑆) ≠ ∅)
3431, 33bitri 275 . . . . . . . . . . 11 ((𝑆𝑥) = ∅ ↔ ¬ (𝑥𝑆) ≠ ∅)
351opncld 22407 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ Top ∧ 𝑥𝐽) → (𝑋𝑥) ∈ (Clsd‘𝐽))
3635adantlr 714 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) → (𝑋𝑥) ∈ (Clsd‘𝐽))
37 reldisj 4415 . . . . . . . . . . . . . . . . 17 (𝑆𝑋 → ((𝑆𝑥) = ∅ ↔ 𝑆 ⊆ (𝑋𝑥)))
3837biimpa 478 . . . . . . . . . . . . . . . 16 ((𝑆𝑋 ∧ (𝑆𝑥) = ∅) → 𝑆 ⊆ (𝑋𝑥))
3938ad4ant24 753 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) ∧ (𝑆𝑥) = ∅) → 𝑆 ⊆ (𝑋𝑥))
401clsss2 22446 . . . . . . . . . . . . . . 15 (((𝑋𝑥) ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ (𝑋𝑥)) → ((cls‘𝐽)‘𝑆) ⊆ (𝑋𝑥))
4136, 39, 40syl2an2r 684 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) ∧ (𝑆𝑥) = ∅) → ((cls‘𝐽)‘𝑆) ⊆ (𝑋𝑥))
4241sseld 3947 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) ∧ (𝑆𝑥) = ∅) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) → 𝑃 ∈ (𝑋𝑥)))
43 eldifn 4091 . . . . . . . . . . . . 13 (𝑃 ∈ (𝑋𝑥) → ¬ 𝑃𝑥)
4442, 43syl6 35 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) ∧ (𝑆𝑥) = ∅) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) → ¬ 𝑃𝑥))
4544con2d 134 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) ∧ (𝑆𝑥) = ∅) → (𝑃𝑥 → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
4634, 45sylan2br 596 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥𝐽) ∧ ¬ (𝑥𝑆) ≠ ∅) → (𝑃𝑥 → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
4746exp31 421 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥𝐽 → (¬ (𝑥𝑆) ≠ ∅ → (𝑃𝑥 → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))))
4847com34 91 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥𝐽 → (𝑃𝑥 → (¬ (𝑥𝑆) ≠ ∅ → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))))
4948imp4a 424 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥𝐽 → ((𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆))))
5049rexlimdv 3147 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆)))
5150imp 408 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ ∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅)) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆))
52513adantl3 1169 . . . 4 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) ∧ ∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅)) → ¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆))
5329, 52impbida 800 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → (¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅)))
54 rexanali 3102 . . 3 (∃𝑥𝐽 (𝑃𝑥 ∧ ¬ (𝑥𝑆) ≠ ∅) ↔ ¬ ∀𝑥𝐽 (𝑃𝑥 → (𝑥𝑆) ≠ ∅))
5553, 54bitrdi 287 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → (¬ 𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ¬ ∀𝑥𝐽 (𝑃𝑥 → (𝑥𝑆) ≠ ∅)))
5655con4bid 317 1 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑥𝐽 (𝑃𝑥 → (𝑥𝑆) ≠ ∅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2940  wral 3061  wrex 3070  cdif 3911  cin 3913  wss 3914  c0 4286   cuni 4869  cfv 6500  Topctop 22265  Clsdccld 22390  clsccl 22392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-top 22266  df-cld 22393  df-ntr 22394  df-cls 22395
This theorem is referenced by:  elcls2  22448  clsndisj  22449  elcls3  22457  neindisj2  22497  islp3  22520  lmcls  22676  1stccnp  22836  txcls  22978  dfac14lem  22991  fclsopn  23388  metdseq0  24240  qndenserrn  44630
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