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Theorem indiscld 20800
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 20711 . . . . 5 {∅, 𝐴} ∈ Top
2 indisuni 20712 . . . . . 6 ( I ‘𝐴) = {∅, 𝐴}
32iscld 20736 . . . . 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 473 . . . . . 6 ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → 𝑥 ⊆ ( I ‘𝐴))
6 dfss4 3841 . . . . . 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 477 . . . . . . 7 ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴})
9 indislem 20709 . . . . . . 7 {∅, ( I ‘𝐴)} = {∅, 𝐴}
108, 9syl6eleqr 2715 . . . . . 6 ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → (( I ‘𝐴) ∖ 𝑥) ∈ {∅, ( I ‘𝐴)})
11 elpri 4173 . . . . . 6 ((( I ‘𝐴) ∖ 𝑥) ∈ {∅, ( I ‘𝐴)} → ((( I ‘𝐴) ∖ 𝑥) = ∅ ∨ (( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴)))
12 difeq2 3705 . . . . . . . . 9 ((( I ‘𝐴) ∖ 𝑥) = ∅ → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = (( I ‘𝐴) ∖ ∅))
13 dif0 3929 . . . . . . . . 9 (( I ‘𝐴) ∖ ∅) = ( I ‘𝐴)
1412, 13syl6eq 2676 . . . . . . . 8 ((( I ‘𝐴) ∖ 𝑥) = ∅ → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = ( I ‘𝐴))
15 fvex 6160 . . . . . . . . . 10 ( I ‘𝐴) ∈ V
1615prid2 4273 . . . . . . . . 9 ( I ‘𝐴) ∈ {∅, ( I ‘𝐴)}
1716, 9eleqtri 2702 . . . . . . . 8 ( I ‘𝐴) ∈ {∅, 𝐴}
1814, 17syl6eqel 2712 . . . . . . 7 ((( I ‘𝐴) ∖ 𝑥) = ∅ → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) ∈ {∅, 𝐴})
19 difeq2 3705 . . . . . . . . 9 ((( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = (( I ‘𝐴) ∖ ( I ‘𝐴)))
20 difid 3927 . . . . . . . . 9 (( I ‘𝐴) ∖ ( I ‘𝐴)) = ∅
2119, 20syl6eq 2676 . . . . . . . 8 ((( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = ∅)
22 0ex 4755 . . . . . . . . 9 ∅ ∈ V
2322prid1 4272 . . . . . . . 8 ∅ ∈ {∅, 𝐴}
2421, 23syl6eqel 2712 . . . . . . 7 ((( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) ∈ {∅, 𝐴})
2518, 24jaoi 394 . . . . . 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 2705 . . . 4 ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → 𝑥 ∈ {∅, 𝐴})
284, 27sylbi 207 . . 3 (𝑥 ∈ (Clsd‘{∅, 𝐴}) → 𝑥 ∈ {∅, 𝐴})
2928ssriv 3592 . 2 (Clsd‘{∅, 𝐴}) ⊆ {∅, 𝐴}
30 0cld 20747 . . . . 5 ({∅, 𝐴} ∈ Top → ∅ ∈ (Clsd‘{∅, 𝐴}))
311, 30ax-mp 5 . . . 4 ∅ ∈ (Clsd‘{∅, 𝐴})
322topcld 20744 . . . . 5 ({∅, 𝐴} ∈ Top → ( I ‘𝐴) ∈ (Clsd‘{∅, 𝐴}))
331, 32ax-mp 5 . . . 4 ( I ‘𝐴) ∈ (Clsd‘{∅, 𝐴})
34 prssi 4326 . . . 4 ((∅ ∈ (Clsd‘{∅, 𝐴}) ∧ ( I ‘𝐴) ∈ (Clsd‘{∅, 𝐴})) → {∅, ( I ‘𝐴)} ⊆ (Clsd‘{∅, 𝐴}))
3531, 33, 34mp2an 707 . . 3 {∅, ( I ‘𝐴)} ⊆ (Clsd‘{∅, 𝐴})
369, 35eqsstr3i 3620 . 2 {∅, 𝐴} ⊆ (Clsd‘{∅, 𝐴})
3729, 36eqssi 3604 1 (Clsd‘{∅, 𝐴}) = {∅, 𝐴}
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 383  wa 384   = wceq 1480  wcel 1992  cdif 3557  wss 3560  c0 3896  {cpr 4155   I cid 4989  cfv 5850  Topctop 20612  Clsdccld 20725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fun 5852  df-fv 5858  df-top 20616  df-topon 20618  df-cld 20728
This theorem is referenced by: (None)
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