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Theorem t1ficld 21911
Description: In a T1 space, finite sets are closed. (Contributed by Mario Carneiro, 25-Dec-2016.)
Hypothesis
Ref Expression
ist0.1 𝑋 = 𝐽
Assertion
Ref Expression
t1ficld ((𝐽 ∈ Fre ∧ 𝐴𝑋𝐴 ∈ Fin) → 𝐴 ∈ (Clsd‘𝐽))

Proof of Theorem t1ficld
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 iunid 4957 . 2 𝑥𝐴 {𝑥} = 𝐴
2 ist0.1 . . . . . 6 𝑋 = 𝐽
32ist1 21905 . . . . 5 (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋 {𝑥} ∈ (Clsd‘𝐽)))
43simplbi 501 . . . 4 (𝐽 ∈ Fre → 𝐽 ∈ Top)
543ad2ant1 1130 . . 3 ((𝐽 ∈ Fre ∧ 𝐴𝑋𝐴 ∈ Fin) → 𝐽 ∈ Top)
6 simp3 1135 . . 3 ((𝐽 ∈ Fre ∧ 𝐴𝑋𝐴 ∈ Fin) → 𝐴 ∈ Fin)
73simprbi 500 . . . . 5 (𝐽 ∈ Fre → ∀𝑥𝑋 {𝑥} ∈ (Clsd‘𝐽))
8 ssralv 4009 . . . . 5 (𝐴𝑋 → (∀𝑥𝑋 {𝑥} ∈ (Clsd‘𝐽) → ∀𝑥𝐴 {𝑥} ∈ (Clsd‘𝐽)))
97, 8mpan9 510 . . . 4 ((𝐽 ∈ Fre ∧ 𝐴𝑋) → ∀𝑥𝐴 {𝑥} ∈ (Clsd‘𝐽))
1093adant3 1129 . . 3 ((𝐽 ∈ Fre ∧ 𝐴𝑋𝐴 ∈ Fin) → ∀𝑥𝐴 {𝑥} ∈ (Clsd‘𝐽))
112iuncld 21629 . . 3 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 {𝑥} ∈ (Clsd‘𝐽)) → 𝑥𝐴 {𝑥} ∈ (Clsd‘𝐽))
125, 6, 10, 11syl3anc 1368 . 2 ((𝐽 ∈ Fre ∧ 𝐴𝑋𝐴 ∈ Fin) → 𝑥𝐴 {𝑥} ∈ (Clsd‘𝐽))
131, 12eqeltrrid 2917 1 ((𝐽 ∈ Fre ∧ 𝐴𝑋𝐴 ∈ Fin) → 𝐴 ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2115  wral 3126  wss 3910  {csn 4540   cuni 4811   ciun 4892  cfv 6328  Fincfn 8484  Topctop 21477  Clsdccld 21600  Frect1 21891
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-iin 4895  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-oadd 8081  df-er 8264  df-en 8485  df-dom 8486  df-fin 8488  df-top 21478  df-cld 21603  df-t1 21898
This theorem is referenced by:  poimirlem30  34973
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