MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  indistopon Structured version   Visualization version   GIF version

Theorem indistopon 20710
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 4339 . . . . 5 (𝑥 ⊆ {∅, 𝐴} ↔ ((𝑥 = ∅ ∨ 𝑥 = {∅}) ∨ (𝑥 = {𝐴} ∨ 𝑥 = {∅, 𝐴})))
2 unieq 4415 . . . . . . . . 9 (𝑥 = ∅ → 𝑥 = ∅)
3 uni0 4436 . . . . . . . . . 10 ∅ = ∅
4 0ex 4755 . . . . . . . . . . 11 ∅ ∈ V
54prid1 4272 . . . . . . . . . 10 ∅ ∈ {∅, 𝐴}
63, 5eqeltri 2700 . . . . . . . . 9 ∅ ∈ {∅, 𝐴}
72, 6syl6eqel 2712 . . . . . . . 8 (𝑥 = ∅ → 𝑥 ∈ {∅, 𝐴})
87a1i 11 . . . . . . 7 (𝐴𝑉 → (𝑥 = ∅ → 𝑥 ∈ {∅, 𝐴}))
9 unieq 4415 . . . . . . . . 9 (𝑥 = {∅} → 𝑥 = {∅})
104unisn 4422 . . . . . . . . . 10 {∅} = ∅
1110, 5eqeltri 2700 . . . . . . . . 9 {∅} ∈ {∅, 𝐴}
129, 11syl6eqel 2712 . . . . . . . 8 (𝑥 = {∅} → 𝑥 ∈ {∅, 𝐴})
1312a1i 11 . . . . . . 7 (𝐴𝑉 → (𝑥 = {∅} → 𝑥 ∈ {∅, 𝐴}))
148, 13jaod 395 . . . . . 6 (𝐴𝑉 → ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 ∈ {∅, 𝐴}))
15 unieq 4415 . . . . . . . . . 10 (𝑥 = {𝐴} → 𝑥 = {𝐴})
16 unisng 4423 . . . . . . . . . 10 (𝐴𝑉 {𝐴} = 𝐴)
1715, 16sylan9eqr 2682 . . . . . . . . 9 ((𝐴𝑉𝑥 = {𝐴}) → 𝑥 = 𝐴)
18 prid2g 4271 . . . . . . . . . 10 (𝐴𝑉𝐴 ∈ {∅, 𝐴})
1918adantr 481 . . . . . . . . 9 ((𝐴𝑉𝑥 = {𝐴}) → 𝐴 ∈ {∅, 𝐴})
2017, 19eqeltrd 2704 . . . . . . . 8 ((𝐴𝑉𝑥 = {𝐴}) → 𝑥 ∈ {∅, 𝐴})
2120ex 450 . . . . . . 7 (𝐴𝑉 → (𝑥 = {𝐴} → 𝑥 ∈ {∅, 𝐴}))
22 unieq 4415 . . . . . . . . . 10 (𝑥 = {∅, 𝐴} → 𝑥 = {∅, 𝐴})
23 uniprg 4421 . . . . . . . . . . . 12 ((∅ ∈ V ∧ 𝐴𝑉) → {∅, 𝐴} = (∅ ∪ 𝐴))
244, 23mpan 705 . . . . . . . . . . 11 (𝐴𝑉 {∅, 𝐴} = (∅ ∪ 𝐴))
25 uncom 3740 . . . . . . . . . . . 12 (∅ ∪ 𝐴) = (𝐴 ∪ ∅)
26 un0 3944 . . . . . . . . . . . 12 (𝐴 ∪ ∅) = 𝐴
2725, 26eqtri 2648 . . . . . . . . . . 11 (∅ ∪ 𝐴) = 𝐴
2824, 27syl6eq 2676 . . . . . . . . . 10 (𝐴𝑉 {∅, 𝐴} = 𝐴)
2922, 28sylan9eqr 2682 . . . . . . . . 9 ((𝐴𝑉𝑥 = {∅, 𝐴}) → 𝑥 = 𝐴)
3018adantr 481 . . . . . . . . 9 ((𝐴𝑉𝑥 = {∅, 𝐴}) → 𝐴 ∈ {∅, 𝐴})
3129, 30eqeltrd 2704 . . . . . . . 8 ((𝐴𝑉𝑥 = {∅, 𝐴}) → 𝑥 ∈ {∅, 𝐴})
3231ex 450 . . . . . . 7 (𝐴𝑉 → (𝑥 = {∅, 𝐴} → 𝑥 ∈ {∅, 𝐴}))
3321, 32jaod 395 . . . . . 6 (𝐴𝑉 → ((𝑥 = {𝐴} ∨ 𝑥 = {∅, 𝐴}) → 𝑥 ∈ {∅, 𝐴}))
3414, 33jaod 395 . . . . 5 (𝐴𝑉 → (((𝑥 = ∅ ∨ 𝑥 = {∅}) ∨ (𝑥 = {𝐴} ∨ 𝑥 = {∅, 𝐴})) → 𝑥 ∈ {∅, 𝐴}))
351, 34syl5bi 232 . . . 4 (𝐴𝑉 → (𝑥 ⊆ {∅, 𝐴} → 𝑥 ∈ {∅, 𝐴}))
3635alrimiv 1857 . . 3 (𝐴𝑉 → ∀𝑥(𝑥 ⊆ {∅, 𝐴} → 𝑥 ∈ {∅, 𝐴}))
37 vex 3194 . . . . . 6 𝑥 ∈ V
3837elpr 4174 . . . . 5 (𝑥 ∈ {∅, 𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐴))
39 vex 3194 . . . . . . . . 9 𝑦 ∈ V
4039elpr 4174 . . . . . . . 8 (𝑦 ∈ {∅, 𝐴} ↔ (𝑦 = ∅ ∨ 𝑦 = 𝐴))
41 simpr 477 . . . . . . . . . . . . . 14 ((𝑥 = ∅ ∧ 𝑦 = ∅) → 𝑦 = ∅)
4241ineq2d 3797 . . . . . . . . . . . . 13 ((𝑥 = ∅ ∧ 𝑦 = ∅) → (𝑥𝑦) = (𝑥 ∩ ∅))
43 in0 3945 . . . . . . . . . . . . 13 (𝑥 ∩ ∅) = ∅
4442, 43syl6eq 2676 . . . . . . . . . . . 12 ((𝑥 = ∅ ∧ 𝑦 = ∅) → (𝑥𝑦) = ∅)
4544, 5syl6eqel 2712 . . . . . . . . . . 11 ((𝑥 = ∅ ∧ 𝑦 = ∅) → (𝑥𝑦) ∈ {∅, 𝐴})
4645a1i 11 . . . . . . . . . 10 (𝐴𝑉 → ((𝑥 = ∅ ∧ 𝑦 = ∅) → (𝑥𝑦) ∈ {∅, 𝐴}))
47 simpr 477 . . . . . . . . . . . . . 14 ((𝑥 = 𝐴𝑦 = ∅) → 𝑦 = ∅)
4847ineq2d 3797 . . . . . . . . . . . . 13 ((𝑥 = 𝐴𝑦 = ∅) → (𝑥𝑦) = (𝑥 ∩ ∅))
4948, 43syl6eq 2676 . . . . . . . . . . . 12 ((𝑥 = 𝐴𝑦 = ∅) → (𝑥𝑦) = ∅)
5049, 5syl6eqel 2712 . . . . . . . . . . 11 ((𝑥 = 𝐴𝑦 = ∅) → (𝑥𝑦) ∈ {∅, 𝐴})
5150a1i 11 . . . . . . . . . 10 (𝐴𝑉 → ((𝑥 = 𝐴𝑦 = ∅) → (𝑥𝑦) ∈ {∅, 𝐴}))
52 simpl 473 . . . . . . . . . . . . . 14 ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → 𝑥 = ∅)
5352ineq1d 3796 . . . . . . . . . . . . 13 ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → (𝑥𝑦) = (∅ ∩ 𝑦))
54 0in 3946 . . . . . . . . . . . . 13 (∅ ∩ 𝑦) = ∅
5553, 54syl6eq 2676 . . . . . . . . . . . 12 ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → (𝑥𝑦) = ∅)
5655, 5syl6eqel 2712 . . . . . . . . . . 11 ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → (𝑥𝑦) ∈ {∅, 𝐴})
5756a1i 11 . . . . . . . . . 10 (𝐴𝑉 → ((𝑥 = ∅ ∧ 𝑦 = 𝐴) → (𝑥𝑦) ∈ {∅, 𝐴}))
58 ineq12 3792 . . . . . . . . . . . . . 14 ((𝑥 = 𝐴𝑦 = 𝐴) → (𝑥𝑦) = (𝐴𝐴))
5958adantl 482 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝑥 = 𝐴𝑦 = 𝐴)) → (𝑥𝑦) = (𝐴𝐴))
60 inidm 3805 . . . . . . . . . . . . 13 (𝐴𝐴) = 𝐴
6159, 60syl6eq 2676 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑥 = 𝐴𝑦 = 𝐴)) → (𝑥𝑦) = 𝐴)
6218adantr 481 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑥 = 𝐴𝑦 = 𝐴)) → 𝐴 ∈ {∅, 𝐴})
6361, 62eqeltrd 2704 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑥 = 𝐴𝑦 = 𝐴)) → (𝑥𝑦) ∈ {∅, 𝐴})
6463ex 450 . . . . . . . . . 10 (𝐴𝑉 → ((𝑥 = 𝐴𝑦 = 𝐴) → (𝑥𝑦) ∈ {∅, 𝐴}))
6546, 51, 57, 64ccased 987 . . . . . . . . 9 (𝐴𝑉 → (((𝑥 = ∅ ∨ 𝑥 = 𝐴) ∧ (𝑦 = ∅ ∨ 𝑦 = 𝐴)) → (𝑥𝑦) ∈ {∅, 𝐴}))
6665expdimp 453 . . . . . . . 8 ((𝐴𝑉 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) → ((𝑦 = ∅ ∨ 𝑦 = 𝐴) → (𝑥𝑦) ∈ {∅, 𝐴}))
6740, 66syl5bi 232 . . . . . . 7 ((𝐴𝑉 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) → (𝑦 ∈ {∅, 𝐴} → (𝑥𝑦) ∈ {∅, 𝐴}))
6867ralrimiv 2964 . . . . . 6 ((𝐴𝑉 ∧ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) → ∀𝑦 ∈ {∅, 𝐴} (𝑥𝑦) ∈ {∅, 𝐴})
6968ex 450 . . . . 5 (𝐴𝑉 → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) → ∀𝑦 ∈ {∅, 𝐴} (𝑥𝑦) ∈ {∅, 𝐴}))
7038, 69syl5bi 232 . . . 4 (𝐴𝑉 → (𝑥 ∈ {∅, 𝐴} → ∀𝑦 ∈ {∅, 𝐴} (𝑥𝑦) ∈ {∅, 𝐴}))
7170ralrimiv 2964 . . 3 (𝐴𝑉 → ∀𝑥 ∈ {∅, 𝐴}∀𝑦 ∈ {∅, 𝐴} (𝑥𝑦) ∈ {∅, 𝐴})
72 prex 4875 . . . 4 {∅, 𝐴} ∈ V
73 istopg 20620 . . . 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 956 . 2 (𝐴𝑉 → {∅, 𝐴} ∈ Top)
7628eqcomd 2632 . 2 (𝐴𝑉𝐴 = {∅, 𝐴})
77 istopon 20635 . 2 ({∅, 𝐴} ∈ (TopOn‘𝐴) ↔ ({∅, 𝐴} ∈ Top ∧ 𝐴 = {∅, 𝐴}))
7875, 76, 77sylanbrc 697 1 (𝐴𝑉 → {∅, 𝐴} ∈ (TopOn‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384  wal 1478   = wceq 1480  wcel 1992  wral 2912  Vcvv 3191  cun 3558  cin 3559  wss 3560  c0 3896  {csn 4153  {cpr 4155   cuni 4407  cfv 5850  Topctop 20612  TopOnctopon 20613
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
This theorem is referenced by:  indistop  20711  indisuni  20712  indistpsx  20719  indistpsALT  20722  indistps2ALT  20723  cnindis  21001  indishmph  21506  indistgp  21809  topdifinf  32821
  Copyright terms: Public domain W3C validator