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Theorem indiscld 20943
Description: The closed sets of an indiscrete topology. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 14-Aug-2015.)
Assertion
Ref Expression
indiscld (Clsd‘{∅, 𝐴}) = {∅, 𝐴}

Proof of Theorem indiscld
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 indistop 20854 . . . . 5 {∅, 𝐴} ∈ Top
2 indisuni 20855 . . . . . 6 ( I ‘𝐴) = {∅, 𝐴}
32iscld 20879 . . . . 5 ({∅, 𝐴} ∈ Top → (𝑥 ∈ (Clsd‘{∅, 𝐴}) ↔ (𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴})))
41, 3ax-mp 5 . . . 4 (𝑥 ∈ (Clsd‘{∅, 𝐴}) ↔ (𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}))
5 simpl 472 . . . . . 6 ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → 𝑥 ⊆ ( I ‘𝐴))
6 dfss4 3891 . . . . . 6 (𝑥 ⊆ ( I ‘𝐴) ↔ (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = 𝑥)
75, 6sylib 208 . . . . 5 ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = 𝑥)
8 simpr 476 . . . . . . 7 ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴})
9 indislem 20852 . . . . . . 7 {∅, ( I ‘𝐴)} = {∅, 𝐴}
108, 9syl6eleqr 2741 . . . . . 6 ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → (( I ‘𝐴) ∖ 𝑥) ∈ {∅, ( I ‘𝐴)})
11 elpri 4230 . . . . . 6 ((( I ‘𝐴) ∖ 𝑥) ∈ {∅, ( I ‘𝐴)} → ((( I ‘𝐴) ∖ 𝑥) = ∅ ∨ (( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴)))
12 difeq2 3755 . . . . . . . . 9 ((( I ‘𝐴) ∖ 𝑥) = ∅ → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = (( I ‘𝐴) ∖ ∅))
13 dif0 3983 . . . . . . . . 9 (( I ‘𝐴) ∖ ∅) = ( I ‘𝐴)
1412, 13syl6eq 2701 . . . . . . . 8 ((( I ‘𝐴) ∖ 𝑥) = ∅ → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = ( I ‘𝐴))
15 fvex 6239 . . . . . . . . . 10 ( I ‘𝐴) ∈ V
1615prid2 4330 . . . . . . . . 9 ( I ‘𝐴) ∈ {∅, ( I ‘𝐴)}
1716, 9eleqtri 2728 . . . . . . . 8 ( I ‘𝐴) ∈ {∅, 𝐴}
1814, 17syl6eqel 2738 . . . . . . 7 ((( I ‘𝐴) ∖ 𝑥) = ∅ → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) ∈ {∅, 𝐴})
19 difeq2 3755 . . . . . . . . 9 ((( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = (( I ‘𝐴) ∖ ( I ‘𝐴)))
20 difid 3981 . . . . . . . . 9 (( I ‘𝐴) ∖ ( I ‘𝐴)) = ∅
2119, 20syl6eq 2701 . . . . . . . 8 ((( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = ∅)
22 0ex 4823 . . . . . . . . 9 ∅ ∈ V
2322prid1 4329 . . . . . . . 8 ∅ ∈ {∅, 𝐴}
2421, 23syl6eqel 2738 . . . . . . 7 ((( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) ∈ {∅, 𝐴})
2518, 24jaoi 393 . . . . . 6 (((( I ‘𝐴) ∖ 𝑥) = ∅ ∨ (( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴)) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) ∈ {∅, 𝐴})
2610, 11, 253syl 18 . . . . 5 ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) ∈ {∅, 𝐴})
277, 26eqeltrrd 2731 . . . 4 ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → 𝑥 ∈ {∅, 𝐴})
284, 27sylbi 207 . . 3 (𝑥 ∈ (Clsd‘{∅, 𝐴}) → 𝑥 ∈ {∅, 𝐴})
2928ssriv 3640 . 2 (Clsd‘{∅, 𝐴}) ⊆ {∅, 𝐴}
30 0cld 20890 . . . . 5 ({∅, 𝐴} ∈ Top → ∅ ∈ (Clsd‘{∅, 𝐴}))
311, 30ax-mp 5 . . . 4 ∅ ∈ (Clsd‘{∅, 𝐴})
322topcld 20887 . . . . 5 ({∅, 𝐴} ∈ Top → ( I ‘𝐴) ∈ (Clsd‘{∅, 𝐴}))
331, 32ax-mp 5 . . . 4 ( I ‘𝐴) ∈ (Clsd‘{∅, 𝐴})
34 prssi 4385 . . . 4 ((∅ ∈ (Clsd‘{∅, 𝐴}) ∧ ( I ‘𝐴) ∈ (Clsd‘{∅, 𝐴})) → {∅, ( I ‘𝐴)} ⊆ (Clsd‘{∅, 𝐴}))
3531, 33, 34mp2an 708 . . 3 {∅, ( I ‘𝐴)} ⊆ (Clsd‘{∅, 𝐴})
369, 35eqsstr3i 3669 . 2 {∅, 𝐴} ⊆ (Clsd‘{∅, 𝐴})
3729, 36eqssi 3652 1 (Clsd‘{∅, 𝐴}) = {∅, 𝐴}
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  cdif 3604  wss 3607  c0 3948  {cpr 4212   I cid 5052  cfv 5926  Topctop 20746  Clsdccld 20868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-top 20747  df-topon 20764  df-cld 20871
This theorem is referenced by: (None)
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