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Theorem indistopon 20615
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 4306 . . . . 5 (𝑥 ⊆ {∅, 𝐴} ↔ ((𝑥 = ∅ ∨ 𝑥 = {∅}) ∨ (𝑥 = {𝐴} ∨ 𝑥 = {∅, 𝐴})))
2 unieq 4380 . . . . . . . . 9 (𝑥 = ∅ → 𝑥 = ∅)
3 uni0 4401 . . . . . . . . . 10 ∅ = ∅
4 0ex 4718 . . . . . . . . . . 11 ∅ ∈ V
54prid1 4241 . . . . . . . . . 10 ∅ ∈ {∅, 𝐴}
63, 5eqeltri 2684 . . . . . . . . 9 ∅ ∈ {∅, 𝐴}
72, 6syl6eqel 2696 . . . . . . . 8 (𝑥 = ∅ → 𝑥 ∈ {∅, 𝐴})
87a1i 11 . . . . . . 7 (𝐴𝑉 → (𝑥 = ∅ → 𝑥 ∈ {∅, 𝐴}))
9 unieq 4380 . . . . . . . . 9 (𝑥 = {∅} → 𝑥 = {∅})
104unisn 4387 . . . . . . . . . 10 {∅} = ∅
1110, 5eqeltri 2684 . . . . . . . . 9 {∅} ∈ {∅, 𝐴}
129, 11syl6eqel 2696 . . . . . . . 8 (𝑥 = {∅} → 𝑥 ∈ {∅, 𝐴})
1312a1i 11 . . . . . . 7 (𝐴𝑉 → (𝑥 = {∅} → 𝑥 ∈ {∅, 𝐴}))
148, 13jaod 394 . . . . . 6 (𝐴𝑉 → ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 ∈ {∅, 𝐴}))
15 unieq 4380 . . . . . . . . . 10 (𝑥 = {𝐴} → 𝑥 = {𝐴})
16 unisng 4388 . . . . . . . . . 10 (𝐴𝑉 {𝐴} = 𝐴)
1715, 16sylan9eqr 2666 . . . . . . . . 9 ((𝐴𝑉𝑥 = {𝐴}) → 𝑥 = 𝐴)
18 prid2g 4240 . . . . . . . . . 10 (𝐴𝑉𝐴 ∈ {∅, 𝐴})
1918adantr 480 . . . . . . . . 9 ((𝐴𝑉𝑥 = {𝐴}) → 𝐴 ∈ {∅, 𝐴})
2017, 19eqeltrd 2688 . . . . . . . 8 ((𝐴𝑉𝑥 = {𝐴}) → 𝑥 ∈ {∅, 𝐴})
2120ex 449 . . . . . . 7 (𝐴𝑉 → (𝑥 = {𝐴} → 𝑥 ∈ {∅, 𝐴}))
22 unieq 4380 . . . . . . . . . 10 (𝑥 = {∅, 𝐴} → 𝑥 = {∅, 𝐴})
23 uniprg 4386 . . . . . . . . . . . 12 ((∅ ∈ V ∧ 𝐴𝑉) → {∅, 𝐴} = (∅ ∪ 𝐴))
244, 23mpan 702 . . . . . . . . . . 11 (𝐴𝑉 {∅, 𝐴} = (∅ ∪ 𝐴))
25 uncom 3719 . . . . . . . . . . . 12 (∅ ∪ 𝐴) = (𝐴 ∪ ∅)
26 un0 3919 . . . . . . . . . . . 12 (𝐴 ∪ ∅) = 𝐴
2725, 26eqtri 2632 . . . . . . . . . . 11 (∅ ∪ 𝐴) = 𝐴
2824, 27syl6eq 2660 . . . . . . . . . 10 (𝐴𝑉 {∅, 𝐴} = 𝐴)
2922, 28sylan9eqr 2666 . . . . . . . . 9 ((𝐴𝑉𝑥 = {∅, 𝐴}) → 𝑥 = 𝐴)
3018adantr 480 . . . . . . . . 9 ((𝐴𝑉𝑥 = {∅, 𝐴}) → 𝐴 ∈ {∅, 𝐴})
3129, 30eqeltrd 2688 . . . . . . . 8 ((𝐴𝑉𝑥 = {∅, 𝐴}) → 𝑥 ∈ {∅, 𝐴})
3231ex 449 . . . . . . 7 (𝐴𝑉 → (𝑥 = {∅, 𝐴} → 𝑥 ∈ {∅, 𝐴}))
3321, 32jaod 394 . . . . . 6 (𝐴𝑉 → ((𝑥 = {𝐴} ∨ 𝑥 = {∅, 𝐴}) → 𝑥 ∈ {∅, 𝐴}))
3414, 33jaod 394 . . . . 5 (𝐴𝑉 → (((𝑥 = ∅ ∨ 𝑥 = {∅}) ∨ (𝑥 = {𝐴} ∨ 𝑥 = {∅, 𝐴})) → 𝑥 ∈ {∅, 𝐴}))
351, 34syl5bi 231 . . . 4 (𝐴𝑉 → (𝑥 ⊆ {∅, 𝐴} → 𝑥 ∈ {∅, 𝐴}))
3635alrimiv 1842 . . 3 (𝐴𝑉 → ∀𝑥(𝑥 ⊆ {∅, 𝐴} → 𝑥 ∈ {∅, 𝐴}))
37 vex 3176 . . . . . 6 𝑥 ∈ V
3837elpr 4146 . . . . 5 (𝑥 ∈ {∅, 𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐴))
39 vex 3176 . . . . . . . . 9 𝑦 ∈ V
4039elpr 4146 . . . . . . . 8 (𝑦 ∈ {∅, 𝐴} ↔ (𝑦 = ∅ ∨ 𝑦 = 𝐴))
41 simpr 476 . . . . . . . . . . . . . 14 ((𝑥 = ∅ ∧ 𝑦 = ∅) → 𝑦 = ∅)
4241ineq2d 3776 . . . . . . . . . . . . 13 ((𝑥 = ∅ ∧ 𝑦 = ∅) → (𝑥𝑦) = (𝑥 ∩ ∅))
43 in0 3920 . . . . . . . . . . . . 13 (𝑥 ∩ ∅) = ∅
4442, 43syl6eq 2660 . . . . . . . . . . . 12 ((𝑥 = ∅ ∧ 𝑦 = ∅) → (𝑥𝑦) = ∅)
4544, 5syl6eqel 2696 . . . . . . . . . . 11 ((𝑥 = ∅ ∧ 𝑦 = ∅) → (𝑥𝑦) ∈ {∅, 𝐴})
4645a1i 11 . . . . . . . . . 10 (𝐴𝑉 → ((𝑥 = ∅ ∧ 𝑦 = ∅) → (𝑥𝑦) ∈ {∅, 𝐴}))
47 simpr 476 . . . . . . . . . . . . . 14 ((𝑥 = 𝐴𝑦 = ∅) → 𝑦 = ∅)
4847ineq2d 3776 . . . . . . . . . . . . 13 ((𝑥 = 𝐴𝑦 = ∅) → (𝑥𝑦) = (𝑥 ∩ ∅))
4948, 43syl6eq 2660 . . . . . . . . . . . 12 ((𝑥 = 𝐴𝑦 = ∅) → (𝑥𝑦) = ∅)
5049, 5syl6eqel 2696 . . . . . . . . . . 11 ((𝑥 = 𝐴𝑦 = ∅) → (𝑥𝑦) ∈ {∅, 𝐴})
5150a1i 11 . . . . . . . . . 10 (𝐴𝑉 → ((𝑥 = 𝐴𝑦 = ∅) → (𝑥𝑦) ∈ {∅, 𝐴}))
52 simpl 472 . . . . . . . . . . . . . 14 ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → 𝑥 = ∅)
5352ineq1d 3775 . . . . . . . . . . . . 13 ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → (𝑥𝑦) = (∅ ∩ 𝑦))
54 0in 3921 . . . . . . . . . . . . 13 (∅ ∩ 𝑦) = ∅
5553, 54syl6eq 2660 . . . . . . . . . . . 12 ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → (𝑥𝑦) = ∅)
5655, 5syl6eqel 2696 . . . . . . . . . . 11 ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → (𝑥𝑦) ∈ {∅, 𝐴})
5756a1i 11 . . . . . . . . . 10 (𝐴𝑉 → ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → (𝑥𝑦) ∈ {∅, 𝐴}))
58 ineq12 3771 . . . . . . . . . . . . . 14 ((𝑥 = 𝐴𝑦 = 𝐴) → (𝑥𝑦) = (𝐴𝐴))
5958adantl 481 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝑥 = 𝐴𝑦 = 𝐴)) → (𝑥𝑦) = (𝐴𝐴))
60 inidm 3784 . . . . . . . . . . . . 13 (𝐴𝐴) = 𝐴
6159, 60syl6eq 2660 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑥 = 𝐴𝑦 = 𝐴)) → (𝑥𝑦) = 𝐴)
6218adantr 480 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑥 = 𝐴𝑦 = 𝐴)) → 𝐴 ∈ {∅, 𝐴})
6361, 62eqeltrd 2688 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑥 = 𝐴𝑦 = 𝐴)) → (𝑥𝑦) ∈ {∅, 𝐴})
6463ex 449 . . . . . . . . . 10 (𝐴𝑉 → ((𝑥 = 𝐴𝑦 = 𝐴) → (𝑥𝑦) ∈ {∅, 𝐴}))
6546, 51, 57, 64ccased 985 . . . . . . . . 9 (𝐴𝑉 → (((𝑥 = ∅ ∨ 𝑥 = 𝐴) ∧ (𝑦 = ∅ ∨ 𝑦 = 𝐴)) → (𝑥𝑦) ∈ {∅, 𝐴}))
6665expdimp 452 . . . . . . . 8 ((𝐴𝑉 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) → ((𝑦 = ∅ ∨ 𝑦 = 𝐴) → (𝑥𝑦) ∈ {∅, 𝐴}))
6740, 66syl5bi 231 . . . . . . 7 ((𝐴𝑉 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) → (𝑦 ∈ {∅, 𝐴} → (𝑥𝑦) ∈ {∅, 𝐴}))
6867ralrimiv 2948 . . . . . 6 ((𝐴𝑉 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) → ∀𝑦 ∈ {∅, 𝐴} (𝑥𝑦) ∈ {∅, 𝐴})
6968ex 449 . . . . 5 (𝐴𝑉 → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) → ∀𝑦 ∈ {∅, 𝐴} (𝑥𝑦) ∈ {∅, 𝐴}))
7038, 69syl5bi 231 . . . 4 (𝐴𝑉 → (𝑥 ∈ {∅, 𝐴} → ∀𝑦 ∈ {∅, 𝐴} (𝑥𝑦) ∈ {∅, 𝐴}))
7170ralrimiv 2948 . . 3 (𝐴𝑉 → ∀𝑥 ∈ {∅, 𝐴}∀𝑦 ∈ {∅, 𝐴} (𝑥𝑦) ∈ {∅, 𝐴})
72 prex 4836 . . . 4 {∅, 𝐴} ∈ V
73 istopg 20525 . . . 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 959 . 2 (𝐴𝑉 → {∅, 𝐴} ∈ Top)
7628eqcomd 2616 . 2 (𝐴𝑉𝐴 = {∅, 𝐴})
77 istopon 20540 . 2 ({∅, 𝐴} ∈ (TopOn‘𝐴) ↔ ({∅, 𝐴} ∈ Top ∧ 𝐴 = {∅, 𝐴}))
7875, 76, 77sylanbrc 695 1 (𝐴𝑉 → {∅, 𝐴} ∈ (TopOn‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wo 382  wa 383  wal 1473   = wceq 1475  wcel 1977  wral 2896  Vcvv 3173  cun 3538  cin 3539  wss 3540  c0 3874  {csn 4125  {cpr 4127   cuni 4372  cfv 5804  Topctop 20517  TopOnctopon 20518
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
This theorem is referenced by:  indistop  20616  indisuni  20617  indistpsx  20624  indistpsALT  20627  indistps2ALT  20628  cnindis  20906  indishmph  21411  indistgp  21714  topdifinf  32373
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