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Theorem indistopon 22351
Description: The indiscrete topology on a set 𝐴. Part of Example 2 in [Munkres] p. 77. (Contributed by Mario Carneiro, 13-Aug-2015.)
Assertion
Ref Expression
indistopon (𝐴𝑉 → {∅, 𝐴} ∈ (TopOn‘𝐴))

Proof of Theorem indistopon
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sspr 4793 . . . . 5 (𝑥 ⊆ {∅, 𝐴} ↔ ((𝑥 = ∅ ∨ 𝑥 = {∅}) ∨ (𝑥 = {𝐴} ∨ 𝑥 = {∅, 𝐴})))
2 unieq 4876 . . . . . . . . 9 (𝑥 = ∅ → 𝑥 = ∅)
3 uni0 4896 . . . . . . . . . 10 ∅ = ∅
4 0ex 5264 . . . . . . . . . . 11 ∅ ∈ V
54prid1 4723 . . . . . . . . . 10 ∅ ∈ {∅, 𝐴}
63, 5eqeltri 2834 . . . . . . . . 9 ∅ ∈ {∅, 𝐴}
72, 6eqeltrdi 2846 . . . . . . . 8 (𝑥 = ∅ → 𝑥 ∈ {∅, 𝐴})
87a1i 11 . . . . . . 7 (𝐴𝑉 → (𝑥 = ∅ → 𝑥 ∈ {∅, 𝐴}))
9 unieq 4876 . . . . . . . . 9 (𝑥 = {∅} → 𝑥 = {∅})
104unisn 4887 . . . . . . . . . 10 {∅} = ∅
1110, 5eqeltri 2834 . . . . . . . . 9 {∅} ∈ {∅, 𝐴}
129, 11eqeltrdi 2846 . . . . . . . 8 (𝑥 = {∅} → 𝑥 ∈ {∅, 𝐴})
1312a1i 11 . . . . . . 7 (𝐴𝑉 → (𝑥 = {∅} → 𝑥 ∈ {∅, 𝐴}))
148, 13jaod 857 . . . . . 6 (𝐴𝑉 → ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 ∈ {∅, 𝐴}))
15 unieq 4876 . . . . . . . . . 10 (𝑥 = {𝐴} → 𝑥 = {𝐴})
16 unisng 4886 . . . . . . . . . 10 (𝐴𝑉 {𝐴} = 𝐴)
1715, 16sylan9eqr 2798 . . . . . . . . 9 ((𝐴𝑉𝑥 = {𝐴}) → 𝑥 = 𝐴)
18 prid2g 4722 . . . . . . . . . 10 (𝐴𝑉𝐴 ∈ {∅, 𝐴})
1918adantr 481 . . . . . . . . 9 ((𝐴𝑉𝑥 = {𝐴}) → 𝐴 ∈ {∅, 𝐴})
2017, 19eqeltrd 2838 . . . . . . . 8 ((𝐴𝑉𝑥 = {𝐴}) → 𝑥 ∈ {∅, 𝐴})
2120ex 413 . . . . . . 7 (𝐴𝑉 → (𝑥 = {𝐴} → 𝑥 ∈ {∅, 𝐴}))
22 unieq 4876 . . . . . . . . . 10 (𝑥 = {∅, 𝐴} → 𝑥 = {∅, 𝐴})
23 uniprg 4882 . . . . . . . . . . . 12 ((∅ ∈ V ∧ 𝐴𝑉) → {∅, 𝐴} = (∅ ∪ 𝐴))
244, 23mpan 688 . . . . . . . . . . 11 (𝐴𝑉 {∅, 𝐴} = (∅ ∪ 𝐴))
25 uncom 4113 . . . . . . . . . . . 12 (∅ ∪ 𝐴) = (𝐴 ∪ ∅)
26 un0 4350 . . . . . . . . . . . 12 (𝐴 ∪ ∅) = 𝐴
2725, 26eqtri 2764 . . . . . . . . . . 11 (∅ ∪ 𝐴) = 𝐴
2824, 27eqtrdi 2792 . . . . . . . . . 10 (𝐴𝑉 {∅, 𝐴} = 𝐴)
2922, 28sylan9eqr 2798 . . . . . . . . 9 ((𝐴𝑉𝑥 = {∅, 𝐴}) → 𝑥 = 𝐴)
3018adantr 481 . . . . . . . . 9 ((𝐴𝑉𝑥 = {∅, 𝐴}) → 𝐴 ∈ {∅, 𝐴})
3129, 30eqeltrd 2838 . . . . . . . 8 ((𝐴𝑉𝑥 = {∅, 𝐴}) → 𝑥 ∈ {∅, 𝐴})
3231ex 413 . . . . . . 7 (𝐴𝑉 → (𝑥 = {∅, 𝐴} → 𝑥 ∈ {∅, 𝐴}))
3321, 32jaod 857 . . . . . 6 (𝐴𝑉 → ((𝑥 = {𝐴} ∨ 𝑥 = {∅, 𝐴}) → 𝑥 ∈ {∅, 𝐴}))
3414, 33jaod 857 . . . . 5 (𝐴𝑉 → (((𝑥 = ∅ ∨ 𝑥 = {∅}) ∨ (𝑥 = {𝐴} ∨ 𝑥 = {∅, 𝐴})) → 𝑥 ∈ {∅, 𝐴}))
351, 34biimtrid 241 . . . 4 (𝐴𝑉 → (𝑥 ⊆ {∅, 𝐴} → 𝑥 ∈ {∅, 𝐴}))
3635alrimiv 1930 . . 3 (𝐴𝑉 → ∀𝑥(𝑥 ⊆ {∅, 𝐴} → 𝑥 ∈ {∅, 𝐴}))
37 vex 3449 . . . . . 6 𝑥 ∈ V
3837elpr 4609 . . . . 5 (𝑥 ∈ {∅, 𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐴))
39 vex 3449 . . . . . . . . 9 𝑦 ∈ V
4039elpr 4609 . . . . . . . 8 (𝑦 ∈ {∅, 𝐴} ↔ (𝑦 = ∅ ∨ 𝑦 = 𝐴))
41 simpr 485 . . . . . . . . . . . . . 14 ((𝑥 = ∅ ∧ 𝑦 = ∅) → 𝑦 = ∅)
4241ineq2d 4172 . . . . . . . . . . . . 13 ((𝑥 = ∅ ∧ 𝑦 = ∅) → (𝑥𝑦) = (𝑥 ∩ ∅))
43 in0 4351 . . . . . . . . . . . . 13 (𝑥 ∩ ∅) = ∅
4442, 43eqtrdi 2792 . . . . . . . . . . . 12 ((𝑥 = ∅ ∧ 𝑦 = ∅) → (𝑥𝑦) = ∅)
4544, 5eqeltrdi 2846 . . . . . . . . . . 11 ((𝑥 = ∅ ∧ 𝑦 = ∅) → (𝑥𝑦) ∈ {∅, 𝐴})
4645a1i 11 . . . . . . . . . 10 (𝐴𝑉 → ((𝑥 = ∅ ∧ 𝑦 = ∅) → (𝑥𝑦) ∈ {∅, 𝐴}))
47 simpr 485 . . . . . . . . . . . . . 14 ((𝑥 = 𝐴𝑦 = ∅) → 𝑦 = ∅)
4847ineq2d 4172 . . . . . . . . . . . . 13 ((𝑥 = 𝐴𝑦 = ∅) → (𝑥𝑦) = (𝑥 ∩ ∅))
4948, 43eqtrdi 2792 . . . . . . . . . . . 12 ((𝑥 = 𝐴𝑦 = ∅) → (𝑥𝑦) = ∅)
5049, 5eqeltrdi 2846 . . . . . . . . . . 11 ((𝑥 = 𝐴𝑦 = ∅) → (𝑥𝑦) ∈ {∅, 𝐴})
5150a1i 11 . . . . . . . . . 10 (𝐴𝑉 → ((𝑥 = 𝐴𝑦 = ∅) → (𝑥𝑦) ∈ {∅, 𝐴}))
52 simpl 483 . . . . . . . . . . . . . 14 ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → 𝑥 = ∅)
5352ineq1d 4171 . . . . . . . . . . . . 13 ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → (𝑥𝑦) = (∅ ∩ 𝑦))
54 0in 4353 . . . . . . . . . . . . 13 (∅ ∩ 𝑦) = ∅
5553, 54eqtrdi 2792 . . . . . . . . . . . 12 ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → (𝑥𝑦) = ∅)
5655, 5eqeltrdi 2846 . . . . . . . . . . 11 ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → (𝑥𝑦) ∈ {∅, 𝐴})
5756a1i 11 . . . . . . . . . 10 (𝐴𝑉 → ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → (𝑥𝑦) ∈ {∅, 𝐴}))
58 ineq12 4167 . . . . . . . . . . . . . 14 ((𝑥 = 𝐴𝑦 = 𝐴) → (𝑥𝑦) = (𝐴𝐴))
5958adantl 482 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝑥 = 𝐴𝑦 = 𝐴)) → (𝑥𝑦) = (𝐴𝐴))
60 inidm 4178 . . . . . . . . . . . . 13 (𝐴𝐴) = 𝐴
6159, 60eqtrdi 2792 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑥 = 𝐴𝑦 = 𝐴)) → (𝑥𝑦) = 𝐴)
6218adantr 481 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑥 = 𝐴𝑦 = 𝐴)) → 𝐴 ∈ {∅, 𝐴})
6361, 62eqeltrd 2838 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑥 = 𝐴𝑦 = 𝐴)) → (𝑥𝑦) ∈ {∅, 𝐴})
6463ex 413 . . . . . . . . . 10 (𝐴𝑉 → ((𝑥 = 𝐴𝑦 = 𝐴) → (𝑥𝑦) ∈ {∅, 𝐴}))
6546, 51, 57, 64ccased 1037 . . . . . . . . 9 (𝐴𝑉 → (((𝑥 = ∅ ∨ 𝑥 = 𝐴) ∧ (𝑦 = ∅ ∨ 𝑦 = 𝐴)) → (𝑥𝑦) ∈ {∅, 𝐴}))
6665expdimp 453 . . . . . . . 8 ((𝐴𝑉 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) → ((𝑦 = ∅ ∨ 𝑦 = 𝐴) → (𝑥𝑦) ∈ {∅, 𝐴}))
6740, 66biimtrid 241 . . . . . . 7 ((𝐴𝑉 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) → (𝑦 ∈ {∅, 𝐴} → (𝑥𝑦) ∈ {∅, 𝐴}))
6867ralrimiv 3142 . . . . . 6 ((𝐴𝑉 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) → ∀𝑦 ∈ {∅, 𝐴} (𝑥𝑦) ∈ {∅, 𝐴})
6968ex 413 . . . . 5 (𝐴𝑉 → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) → ∀𝑦 ∈ {∅, 𝐴} (𝑥𝑦) ∈ {∅, 𝐴}))
7038, 69biimtrid 241 . . . 4 (𝐴𝑉 → (𝑥 ∈ {∅, 𝐴} → ∀𝑦 ∈ {∅, 𝐴} (𝑥𝑦) ∈ {∅, 𝐴}))
7170ralrimiv 3142 . . 3 (𝐴𝑉 → ∀𝑥 ∈ {∅, 𝐴}∀𝑦 ∈ {∅, 𝐴} (𝑥𝑦) ∈ {∅, 𝐴})
72 prex 5389 . . . 4 {∅, 𝐴} ∈ V
73 istopg 22244 . . . 4 ({∅, 𝐴} ∈ V → ({∅, 𝐴} ∈ Top ↔ (∀𝑥(𝑥 ⊆ {∅, 𝐴} → 𝑥 ∈ {∅, 𝐴}) ∧ ∀𝑥 ∈ {∅, 𝐴}∀𝑦 ∈ {∅, 𝐴} (𝑥𝑦) ∈ {∅, 𝐴})))
7472, 73mp1i 13 . . 3 (𝐴𝑉 → ({∅, 𝐴} ∈ Top ↔ (∀𝑥(𝑥 ⊆ {∅, 𝐴} → 𝑥 ∈ {∅, 𝐴}) ∧ ∀𝑥 ∈ {∅, 𝐴}∀𝑦 ∈ {∅, 𝐴} (𝑥𝑦) ∈ {∅, 𝐴})))
7536, 71, 74mpbir2and 711 . 2 (𝐴𝑉 → {∅, 𝐴} ∈ Top)
7628eqcomd 2742 . 2 (𝐴𝑉𝐴 = {∅, 𝐴})
77 istopon 22261 . 2 ({∅, 𝐴} ∈ (TopOn‘𝐴) ↔ ({∅, 𝐴} ∈ Top ∧ 𝐴 = {∅, 𝐴}))
7875, 76, 77sylanbrc 583 1 (𝐴𝑉 → {∅, 𝐴} ∈ (TopOn‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845  wal 1539   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  cun 3908  cin 3909  wss 3910  c0 4282  {csn 4586  {cpr 4588   cuni 4865  cfv 6496  Topctop 22242  TopOnctopon 22259
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-iota 6448  df-fun 6498  df-fv 6504  df-top 22243  df-topon 22260
This theorem is referenced by:  indistop  22352  indisuni  22353  indistpsx  22360  indistpsALT  22363  indistpsALTOLD  22364  indistps2ALT  22365  cnindis  22643  indishmph  23149  indistgp  23451  topdifinf  35820
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