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Theorem indistopon 20853
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 4398 . . . . 5 (𝑥 ⊆ {∅, 𝐴} ↔ ((𝑥 = ∅ ∨ 𝑥 = {∅}) ∨ (𝑥 = {𝐴} ∨ 𝑥 = {∅, 𝐴})))
2 unieq 4476 . . . . . . . . 9 (𝑥 = ∅ → 𝑥 = ∅)
3 uni0 4497 . . . . . . . . . 10 ∅ = ∅
4 0ex 4823 . . . . . . . . . . 11 ∅ ∈ V
54prid1 4329 . . . . . . . . . 10 ∅ ∈ {∅, 𝐴}
63, 5eqeltri 2726 . . . . . . . . 9 ∅ ∈ {∅, 𝐴}
72, 6syl6eqel 2738 . . . . . . . 8 (𝑥 = ∅ → 𝑥 ∈ {∅, 𝐴})
87a1i 11 . . . . . . 7 (𝐴𝑉 → (𝑥 = ∅ → 𝑥 ∈ {∅, 𝐴}))
9 unieq 4476 . . . . . . . . 9 (𝑥 = {∅} → 𝑥 = {∅})
104unisn 4483 . . . . . . . . . 10 {∅} = ∅
1110, 5eqeltri 2726 . . . . . . . . 9 {∅} ∈ {∅, 𝐴}
129, 11syl6eqel 2738 . . . . . . . 8 (𝑥 = {∅} → 𝑥 ∈ {∅, 𝐴})
1312a1i 11 . . . . . . 7 (𝐴𝑉 → (𝑥 = {∅} → 𝑥 ∈ {∅, 𝐴}))
148, 13jaod 394 . . . . . 6 (𝐴𝑉 → ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 ∈ {∅, 𝐴}))
15 unieq 4476 . . . . . . . . . 10 (𝑥 = {𝐴} → 𝑥 = {𝐴})
16 unisng 4484 . . . . . . . . . 10 (𝐴𝑉 {𝐴} = 𝐴)
1715, 16sylan9eqr 2707 . . . . . . . . 9 ((𝐴𝑉𝑥 = {𝐴}) → 𝑥 = 𝐴)
18 prid2g 4328 . . . . . . . . . 10 (𝐴𝑉𝐴 ∈ {∅, 𝐴})
1918adantr 480 . . . . . . . . 9 ((𝐴𝑉𝑥 = {𝐴}) → 𝐴 ∈ {∅, 𝐴})
2017, 19eqeltrd 2730 . . . . . . . 8 ((𝐴𝑉𝑥 = {𝐴}) → 𝑥 ∈ {∅, 𝐴})
2120ex 449 . . . . . . 7 (𝐴𝑉 → (𝑥 = {𝐴} → 𝑥 ∈ {∅, 𝐴}))
22 unieq 4476 . . . . . . . . . 10 (𝑥 = {∅, 𝐴} → 𝑥 = {∅, 𝐴})
23 uniprg 4482 . . . . . . . . . . . 12 ((∅ ∈ V ∧ 𝐴𝑉) → {∅, 𝐴} = (∅ ∪ 𝐴))
244, 23mpan 706 . . . . . . . . . . 11 (𝐴𝑉 {∅, 𝐴} = (∅ ∪ 𝐴))
25 uncom 3790 . . . . . . . . . . . 12 (∅ ∪ 𝐴) = (𝐴 ∪ ∅)
26 un0 4000 . . . . . . . . . . . 12 (𝐴 ∪ ∅) = 𝐴
2725, 26eqtri 2673 . . . . . . . . . . 11 (∅ ∪ 𝐴) = 𝐴
2824, 27syl6eq 2701 . . . . . . . . . 10 (𝐴𝑉 {∅, 𝐴} = 𝐴)
2922, 28sylan9eqr 2707 . . . . . . . . 9 ((𝐴𝑉𝑥 = {∅, 𝐴}) → 𝑥 = 𝐴)
3018adantr 480 . . . . . . . . 9 ((𝐴𝑉𝑥 = {∅, 𝐴}) → 𝐴 ∈ {∅, 𝐴})
3129, 30eqeltrd 2730 . . . . . . . 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 232 . . . 4 (𝐴𝑉 → (𝑥 ⊆ {∅, 𝐴} → 𝑥 ∈ {∅, 𝐴}))
3635alrimiv 1895 . . 3 (𝐴𝑉 → ∀𝑥(𝑥 ⊆ {∅, 𝐴} → 𝑥 ∈ {∅, 𝐴}))
37 vex 3234 . . . . . 6 𝑥 ∈ V
3837elpr 4231 . . . . 5 (𝑥 ∈ {∅, 𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐴))
39 vex 3234 . . . . . . . . 9 𝑦 ∈ V
4039elpr 4231 . . . . . . . 8 (𝑦 ∈ {∅, 𝐴} ↔ (𝑦 = ∅ ∨ 𝑦 = 𝐴))
41 simpr 476 . . . . . . . . . . . . . 14 ((𝑥 = ∅ ∧ 𝑦 = ∅) → 𝑦 = ∅)
4241ineq2d 3847 . . . . . . . . . . . . 13 ((𝑥 = ∅ ∧ 𝑦 = ∅) → (𝑥𝑦) = (𝑥 ∩ ∅))
43 in0 4001 . . . . . . . . . . . . 13 (𝑥 ∩ ∅) = ∅
4442, 43syl6eq 2701 . . . . . . . . . . . 12 ((𝑥 = ∅ ∧ 𝑦 = ∅) → (𝑥𝑦) = ∅)
4544, 5syl6eqel 2738 . . . . . . . . . . 11 ((𝑥 = ∅ ∧ 𝑦 = ∅) → (𝑥𝑦) ∈ {∅, 𝐴})
4645a1i 11 . . . . . . . . . 10 (𝐴𝑉 → ((𝑥 = ∅ ∧ 𝑦 = ∅) → (𝑥𝑦) ∈ {∅, 𝐴}))
47 simpr 476 . . . . . . . . . . . . . 14 ((𝑥 = 𝐴𝑦 = ∅) → 𝑦 = ∅)
4847ineq2d 3847 . . . . . . . . . . . . 13 ((𝑥 = 𝐴𝑦 = ∅) → (𝑥𝑦) = (𝑥 ∩ ∅))
4948, 43syl6eq 2701 . . . . . . . . . . . 12 ((𝑥 = 𝐴𝑦 = ∅) → (𝑥𝑦) = ∅)
5049, 5syl6eqel 2738 . . . . . . . . . . 11 ((𝑥 = 𝐴𝑦 = ∅) → (𝑥𝑦) ∈ {∅, 𝐴})
5150a1i 11 . . . . . . . . . 10 (𝐴𝑉 → ((𝑥 = 𝐴𝑦 = ∅) → (𝑥𝑦) ∈ {∅, 𝐴}))
52 simpl 472 . . . . . . . . . . . . . 14 ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → 𝑥 = ∅)
5352ineq1d 3846 . . . . . . . . . . . . 13 ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → (𝑥𝑦) = (∅ ∩ 𝑦))
54 0in 4002 . . . . . . . . . . . . 13 (∅ ∩ 𝑦) = ∅
5553, 54syl6eq 2701 . . . . . . . . . . . 12 ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → (𝑥𝑦) = ∅)
5655, 5syl6eqel 2738 . . . . . . . . . . 11 ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → (𝑥𝑦) ∈ {∅, 𝐴})
5756a1i 11 . . . . . . . . . 10 (𝐴𝑉 → ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → (𝑥𝑦) ∈ {∅, 𝐴}))
58 ineq12 3842 . . . . . . . . . . . . . 14 ((𝑥 = 𝐴𝑦 = 𝐴) → (𝑥𝑦) = (𝐴𝐴))
5958adantl 481 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝑥 = 𝐴𝑦 = 𝐴)) → (𝑥𝑦) = (𝐴𝐴))
60 inidm 3855 . . . . . . . . . . . . 13 (𝐴𝐴) = 𝐴
6159, 60syl6eq 2701 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑥 = 𝐴𝑦 = 𝐴)) → (𝑥𝑦) = 𝐴)
6218adantr 480 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑥 = 𝐴𝑦 = 𝐴)) → 𝐴 ∈ {∅, 𝐴})
6361, 62eqeltrd 2730 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑥 = 𝐴𝑦 = 𝐴)) → (𝑥𝑦) ∈ {∅, 𝐴})
6463ex 449 . . . . . . . . . 10 (𝐴𝑉 → ((𝑥 = 𝐴𝑦 = 𝐴) → (𝑥𝑦) ∈ {∅, 𝐴}))
6546, 51, 57, 64ccased 1007 . . . . . . . . 9 (𝐴𝑉 → (((𝑥 = ∅ ∨ 𝑥 = 𝐴) ∧ (𝑦 = ∅ ∨ 𝑦 = 𝐴)) → (𝑥𝑦) ∈ {∅, 𝐴}))
6665expdimp 452 . . . . . . . 8 ((𝐴𝑉 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) → ((𝑦 = ∅ ∨ 𝑦 = 𝐴) → (𝑥𝑦) ∈ {∅, 𝐴}))
6740, 66syl5bi 232 . . . . . . 7 ((𝐴𝑉 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) → (𝑦 ∈ {∅, 𝐴} → (𝑥𝑦) ∈ {∅, 𝐴}))
6867ralrimiv 2994 . . . . . 6 ((𝐴𝑉 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) → ∀𝑦 ∈ {∅, 𝐴} (𝑥𝑦) ∈ {∅, 𝐴})
6968ex 449 . . . . 5 (𝐴𝑉 → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) → ∀𝑦 ∈ {∅, 𝐴} (𝑥𝑦) ∈ {∅, 𝐴}))
7038, 69syl5bi 232 . . . 4 (𝐴𝑉 → (𝑥 ∈ {∅, 𝐴} → ∀𝑦 ∈ {∅, 𝐴} (𝑥𝑦) ∈ {∅, 𝐴}))
7170ralrimiv 2994 . . 3 (𝐴𝑉 → ∀𝑥 ∈ {∅, 𝐴}∀𝑦 ∈ {∅, 𝐴} (𝑥𝑦) ∈ {∅, 𝐴})
72 prex 4939 . . . 4 {∅, 𝐴} ∈ V
73 istopg 20748 . . . 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 977 . 2 (𝐴𝑉 → {∅, 𝐴} ∈ Top)
7628eqcomd 2657 . 2 (𝐴𝑉𝐴 = {∅, 𝐴})
77 istopon 20765 . 2 ({∅, 𝐴} ∈ (TopOn‘𝐴) ↔ ({∅, 𝐴} ∈ Top ∧ 𝐴 = {∅, 𝐴}))
7875, 76, 77sylanbrc 699 1 (𝐴𝑉 → {∅, 𝐴} ∈ (TopOn‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  wal 1521   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  cun 3605  cin 3606  wss 3607  c0 3948  {csn 4210  {cpr 4212   cuni 4468  cfv 5926  Topctop 20746  TopOnctopon 20763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-top 20747  df-topon 20764
This theorem is referenced by:  indistop  20854  indisuni  20855  indistpsx  20862  indistpsALT  20865  indistps2ALT  20866  cnindis  21144  indishmph  21649  indistgp  21951  topdifinf  33327
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