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Theorem indiscld 22595
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 22505 . . . . 5 {∅, 𝐴} ∈ Top
2 indisuni 22506 . . . . . 6 ( I ‘𝐴) = {∅, 𝐴}
32iscld 22531 . . . . 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 484 . . . . . 6 ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → 𝑥 ⊆ ( I ‘𝐴))
6 dfss4 4259 . . . . . 6 (𝑥 ⊆ ( I ‘𝐴) ↔ (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = 𝑥)
75, 6sylib 217 . . . . 5 ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = 𝑥)
8 simpr 486 . . . . . . 7 ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴})
9 indislem 22503 . . . . . . 7 {∅, ( I ‘𝐴)} = {∅, 𝐴}
108, 9eleqtrrdi 2845 . . . . . 6 ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → (( I ‘𝐴) ∖ 𝑥) ∈ {∅, ( I ‘𝐴)})
11 elpri 4651 . . . . . 6 ((( I ‘𝐴) ∖ 𝑥) ∈ {∅, ( I ‘𝐴)} → ((( I ‘𝐴) ∖ 𝑥) = ∅ ∨ (( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴)))
12 difeq2 4117 . . . . . . . . 9 ((( I ‘𝐴) ∖ 𝑥) = ∅ → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = (( I ‘𝐴) ∖ ∅))
13 dif0 4373 . . . . . . . . 9 (( I ‘𝐴) ∖ ∅) = ( I ‘𝐴)
1412, 13eqtrdi 2789 . . . . . . . 8 ((( I ‘𝐴) ∖ 𝑥) = ∅ → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = ( I ‘𝐴))
15 fvex 6905 . . . . . . . . . 10 ( I ‘𝐴) ∈ V
1615prid2 4768 . . . . . . . . 9 ( I ‘𝐴) ∈ {∅, ( I ‘𝐴)}
1716, 9eleqtri 2832 . . . . . . . 8 ( I ‘𝐴) ∈ {∅, 𝐴}
1814, 17eqeltrdi 2842 . . . . . . 7 ((( I ‘𝐴) ∖ 𝑥) = ∅ → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) ∈ {∅, 𝐴})
19 difeq2 4117 . . . . . . . . 9 ((( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = (( I ‘𝐴) ∖ ( I ‘𝐴)))
20 difid 4371 . . . . . . . . 9 (( I ‘𝐴) ∖ ( I ‘𝐴)) = ∅
2119, 20eqtrdi 2789 . . . . . . . 8 ((( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) = ∅)
22 0ex 5308 . . . . . . . . 9 ∅ ∈ V
2322prid1 4767 . . . . . . . 8 ∅ ∈ {∅, 𝐴}
2421, 23eqeltrdi 2842 . . . . . . 7 ((( I ‘𝐴) ∖ 𝑥) = ( I ‘𝐴) → (( I ‘𝐴) ∖ (( I ‘𝐴) ∖ 𝑥)) ∈ {∅, 𝐴})
2518, 24jaoi 856 . . . . . 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 2835 . . . 4 ((𝑥 ⊆ ( I ‘𝐴) ∧ (( I ‘𝐴) ∖ 𝑥) ∈ {∅, 𝐴}) → 𝑥 ∈ {∅, 𝐴})
284, 27sylbi 216 . . 3 (𝑥 ∈ (Clsd‘{∅, 𝐴}) → 𝑥 ∈ {∅, 𝐴})
2928ssriv 3987 . 2 (Clsd‘{∅, 𝐴}) ⊆ {∅, 𝐴}
30 0cld 22542 . . . . 5 ({∅, 𝐴} ∈ Top → ∅ ∈ (Clsd‘{∅, 𝐴}))
311, 30ax-mp 5 . . . 4 ∅ ∈ (Clsd‘{∅, 𝐴})
322topcld 22539 . . . . 5 ({∅, 𝐴} ∈ Top → ( I ‘𝐴) ∈ (Clsd‘{∅, 𝐴}))
331, 32ax-mp 5 . . . 4 ( I ‘𝐴) ∈ (Clsd‘{∅, 𝐴})
34 prssi 4825 . . . 4 ((∅ ∈ (Clsd‘{∅, 𝐴}) ∧ ( I ‘𝐴) ∈ (Clsd‘{∅, 𝐴})) → {∅, ( I ‘𝐴)} ⊆ (Clsd‘{∅, 𝐴}))
3531, 33, 34mp2an 691 . . 3 {∅, ( I ‘𝐴)} ⊆ (Clsd‘{∅, 𝐴})
369, 35eqsstrri 4018 . 2 {∅, 𝐴} ⊆ (Clsd‘{∅, 𝐴})
3729, 36eqssi 3999 1 (Clsd‘{∅, 𝐴}) = {∅, 𝐴}
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wo 846   = wceq 1542  wcel 2107  cdif 3946  wss 3949  c0 4323  {cpr 4631   I cid 5574  cfv 6544  Topctop 22395  Clsdccld 22520
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-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-top 22396  df-topon 22413  df-cld 22523
This theorem is referenced by: (None)
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