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Theorem indiscld 20705
 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 20616 . . . . 5 {∅, 𝐴} ∈ Top
2 indisuni 20617 . . . . . 6 ( I ‘𝐴) = {∅, 𝐴}
32iscld 20641 . . . . 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 3820 . . . . . 6 (𝑥 ⊆ ( I ‘𝐴) ↔ (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = 𝑥)
75, 6sylib 207 . . . . 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 20614 . . . . . . 7 {∅, ( I ‘𝐴)} = {∅, 𝐴}
108, 9syl6eleqr 2699 . . . . . 6 ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → (( I ‘𝐴) ∖ 𝑥) ∈ {∅, ( I ‘𝐴)})
11 elpri 4145 . . . . . 6 ((( I ‘𝐴) ∖ 𝑥) ∈ {∅, ( I ‘𝐴)} → ((( I ‘𝐴) ∖ 𝑥) = ∅ ∨ (( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴)))
12 difeq2 3684 . . . . . . . . 9 ((( I ‘𝐴) ∖ 𝑥) = ∅ → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = (( I ‘𝐴) ∖ ∅))
13 dif0 3904 . . . . . . . . 9 (( I ‘𝐴) ∖ ∅) = ( I ‘𝐴)
1412, 13syl6eq 2660 . . . . . . . 8 ((( I ‘𝐴) ∖ 𝑥) = ∅ → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = ( I ‘𝐴))
15 fvex 6113 . . . . . . . . . 10 ( I ‘𝐴) ∈ V
1615prid2 4242 . . . . . . . . 9 ( I ‘𝐴) ∈ {∅, ( I ‘𝐴)}
1716, 9eleqtri 2686 . . . . . . . 8 ( I ‘𝐴) ∈ {∅, 𝐴}
1814, 17syl6eqel 2696 . . . . . . 7 ((( I ‘𝐴) ∖ 𝑥) = ∅ → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) ∈ {∅, 𝐴})
19 difeq2 3684 . . . . . . . . 9 ((( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = (( I ‘𝐴) ∖ ( I ‘𝐴)))
20 difid 3902 . . . . . . . . 9 (( I ‘𝐴) ∖ ( I ‘𝐴)) = ∅
2119, 20syl6eq 2660 . . . . . . . 8 ((( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = ∅)
22 0ex 4718 . . . . . . . . 9 ∅ ∈ V
2322prid1 4241 . . . . . . . 8 ∅ ∈ {∅, 𝐴}
2421, 23syl6eqel 2696 . . . . . . 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 2689 . . . 4 ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → 𝑥 ∈ {∅, 𝐴})
284, 27sylbi 206 . . 3 (𝑥 ∈ (Clsd‘{∅, 𝐴}) → 𝑥 ∈ {∅, 𝐴})
2928ssriv 3572 . 2 (Clsd‘{∅, 𝐴}) ⊆ {∅, 𝐴}
30 0cld 20652 . . . . 5 ({∅, 𝐴} ∈ Top → ∅ ∈ (Clsd‘{∅, 𝐴}))
311, 30ax-mp 5 . . . 4 ∅ ∈ (Clsd‘{∅, 𝐴})
322topcld 20649 . . . . 5 ({∅, 𝐴} ∈ Top → ( I ‘𝐴) ∈ (Clsd‘{∅, 𝐴}))
331, 32ax-mp 5 . . . 4 ( I ‘𝐴) ∈ (Clsd‘{∅, 𝐴})
34 prssi 4293 . . . 4 ((∅ ∈ (Clsd‘{∅, 𝐴}) ∧ ( I ‘𝐴) ∈ (Clsd‘{∅, 𝐴})) → {∅, ( I ‘𝐴)} ⊆ (Clsd‘{∅, 𝐴}))
3531, 33, 34mp2an 704 . . 3 {∅, ( I ‘𝐴)} ⊆ (Clsd‘{∅, 𝐴})
369, 35eqsstr3i 3599 . 2 {∅, 𝐴} ⊆ (Clsd‘{∅, 𝐴})
3729, 36eqssi 3584 1 (Clsd‘{∅, 𝐴}) = {∅, 𝐴}
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  {cpr 4127   I cid 4948  ‘cfv 5804  Topctop 20517  Clsdccld 20630 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-top 20521  df-topon 20523  df-cld 20633 This theorem is referenced by: (None)
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