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Theorem indistopon 22367
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 (𝐴 ∈ 𝑉 β†’ {βˆ…, 𝐴} ∈ (TopOnβ€˜π΄))

Proof of Theorem indistopon
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sspr 4794 . . . . 5 (π‘₯ βŠ† {βˆ…, 𝐴} ↔ ((π‘₯ = βˆ… ∨ π‘₯ = {βˆ…}) ∨ (π‘₯ = {𝐴} ∨ π‘₯ = {βˆ…, 𝐴})))
2 unieq 4877 . . . . . . . . 9 (π‘₯ = βˆ… β†’ βˆͺ π‘₯ = βˆͺ βˆ…)
3 uni0 4897 . . . . . . . . . 10 βˆͺ βˆ… = βˆ…
4 0ex 5265 . . . . . . . . . . 11 βˆ… ∈ V
54prid1 4724 . . . . . . . . . 10 βˆ… ∈ {βˆ…, 𝐴}
63, 5eqeltri 2830 . . . . . . . . 9 βˆͺ βˆ… ∈ {βˆ…, 𝐴}
72, 6eqeltrdi 2842 . . . . . . . 8 (π‘₯ = βˆ… β†’ βˆͺ π‘₯ ∈ {βˆ…, 𝐴})
87a1i 11 . . . . . . 7 (𝐴 ∈ 𝑉 β†’ (π‘₯ = βˆ… β†’ βˆͺ π‘₯ ∈ {βˆ…, 𝐴}))
9 unieq 4877 . . . . . . . . 9 (π‘₯ = {βˆ…} β†’ βˆͺ π‘₯ = βˆͺ {βˆ…})
104unisn 4888 . . . . . . . . . 10 βˆͺ {βˆ…} = βˆ…
1110, 5eqeltri 2830 . . . . . . . . 9 βˆͺ {βˆ…} ∈ {βˆ…, 𝐴}
129, 11eqeltrdi 2842 . . . . . . . 8 (π‘₯ = {βˆ…} β†’ βˆͺ π‘₯ ∈ {βˆ…, 𝐴})
1312a1i 11 . . . . . . 7 (𝐴 ∈ 𝑉 β†’ (π‘₯ = {βˆ…} β†’ βˆͺ π‘₯ ∈ {βˆ…, 𝐴}))
148, 13jaod 858 . . . . . 6 (𝐴 ∈ 𝑉 β†’ ((π‘₯ = βˆ… ∨ π‘₯ = {βˆ…}) β†’ βˆͺ π‘₯ ∈ {βˆ…, 𝐴}))
15 unieq 4877 . . . . . . . . . 10 (π‘₯ = {𝐴} β†’ βˆͺ π‘₯ = βˆͺ {𝐴})
16 unisng 4887 . . . . . . . . . 10 (𝐴 ∈ 𝑉 β†’ βˆͺ {𝐴} = 𝐴)
1715, 16sylan9eqr 2795 . . . . . . . . 9 ((𝐴 ∈ 𝑉 ∧ π‘₯ = {𝐴}) β†’ βˆͺ π‘₯ = 𝐴)
18 prid2g 4723 . . . . . . . . . 10 (𝐴 ∈ 𝑉 β†’ 𝐴 ∈ {βˆ…, 𝐴})
1918adantr 482 . . . . . . . . 9 ((𝐴 ∈ 𝑉 ∧ π‘₯ = {𝐴}) β†’ 𝐴 ∈ {βˆ…, 𝐴})
2017, 19eqeltrd 2834 . . . . . . . 8 ((𝐴 ∈ 𝑉 ∧ π‘₯ = {𝐴}) β†’ βˆͺ π‘₯ ∈ {βˆ…, 𝐴})
2120ex 414 . . . . . . 7 (𝐴 ∈ 𝑉 β†’ (π‘₯ = {𝐴} β†’ βˆͺ π‘₯ ∈ {βˆ…, 𝐴}))
22 unieq 4877 . . . . . . . . . 10 (π‘₯ = {βˆ…, 𝐴} β†’ βˆͺ π‘₯ = βˆͺ {βˆ…, 𝐴})
23 uniprg 4883 . . . . . . . . . . . 12 ((βˆ… ∈ V ∧ 𝐴 ∈ 𝑉) β†’ βˆͺ {βˆ…, 𝐴} = (βˆ… βˆͺ 𝐴))
244, 23mpan 689 . . . . . . . . . . 11 (𝐴 ∈ 𝑉 β†’ βˆͺ {βˆ…, 𝐴} = (βˆ… βˆͺ 𝐴))
25 uncom 4114 . . . . . . . . . . . 12 (βˆ… βˆͺ 𝐴) = (𝐴 βˆͺ βˆ…)
26 un0 4351 . . . . . . . . . . . 12 (𝐴 βˆͺ βˆ…) = 𝐴
2725, 26eqtri 2761 . . . . . . . . . . 11 (βˆ… βˆͺ 𝐴) = 𝐴
2824, 27eqtrdi 2789 . . . . . . . . . 10 (𝐴 ∈ 𝑉 β†’ βˆͺ {βˆ…, 𝐴} = 𝐴)
2922, 28sylan9eqr 2795 . . . . . . . . 9 ((𝐴 ∈ 𝑉 ∧ π‘₯ = {βˆ…, 𝐴}) β†’ βˆͺ π‘₯ = 𝐴)
3018adantr 482 . . . . . . . . 9 ((𝐴 ∈ 𝑉 ∧ π‘₯ = {βˆ…, 𝐴}) β†’ 𝐴 ∈ {βˆ…, 𝐴})
3129, 30eqeltrd 2834 . . . . . . . 8 ((𝐴 ∈ 𝑉 ∧ π‘₯ = {βˆ…, 𝐴}) β†’ βˆͺ π‘₯ ∈ {βˆ…, 𝐴})
3231ex 414 . . . . . . 7 (𝐴 ∈ 𝑉 β†’ (π‘₯ = {βˆ…, 𝐴} β†’ βˆͺ π‘₯ ∈ {βˆ…, 𝐴}))
3321, 32jaod 858 . . . . . 6 (𝐴 ∈ 𝑉 β†’ ((π‘₯ = {𝐴} ∨ π‘₯ = {βˆ…, 𝐴}) β†’ βˆͺ π‘₯ ∈ {βˆ…, 𝐴}))
3414, 33jaod 858 . . . . 5 (𝐴 ∈ 𝑉 β†’ (((π‘₯ = βˆ… ∨ π‘₯ = {βˆ…}) ∨ (π‘₯ = {𝐴} ∨ π‘₯ = {βˆ…, 𝐴})) β†’ βˆͺ π‘₯ ∈ {βˆ…, 𝐴}))
351, 34biimtrid 241 . . . 4 (𝐴 ∈ 𝑉 β†’ (π‘₯ βŠ† {βˆ…, 𝐴} β†’ βˆͺ π‘₯ ∈ {βˆ…, 𝐴}))
3635alrimiv 1931 . . 3 (𝐴 ∈ 𝑉 β†’ βˆ€π‘₯(π‘₯ βŠ† {βˆ…, 𝐴} β†’ βˆͺ π‘₯ ∈ {βˆ…, 𝐴}))
37 vex 3448 . . . . . 6 π‘₯ ∈ V
3837elpr 4610 . . . . 5 (π‘₯ ∈ {βˆ…, 𝐴} ↔ (π‘₯ = βˆ… ∨ π‘₯ = 𝐴))
39 vex 3448 . . . . . . . . 9 𝑦 ∈ V
4039elpr 4610 . . . . . . . 8 (𝑦 ∈ {βˆ…, 𝐴} ↔ (𝑦 = βˆ… ∨ 𝑦 = 𝐴))
41 simpr 486 . . . . . . . . . . . . . 14 ((π‘₯ = βˆ… ∧ 𝑦 = βˆ…) β†’ 𝑦 = βˆ…)
4241ineq2d 4173 . . . . . . . . . . . . 13 ((π‘₯ = βˆ… ∧ 𝑦 = βˆ…) β†’ (π‘₯ ∩ 𝑦) = (π‘₯ ∩ βˆ…))
43 in0 4352 . . . . . . . . . . . . 13 (π‘₯ ∩ βˆ…) = βˆ…
4442, 43eqtrdi 2789 . . . . . . . . . . . 12 ((π‘₯ = βˆ… ∧ 𝑦 = βˆ…) β†’ (π‘₯ ∩ 𝑦) = βˆ…)
4544, 5eqeltrdi 2842 . . . . . . . . . . 11 ((π‘₯ = βˆ… ∧ 𝑦 = βˆ…) β†’ (π‘₯ ∩ 𝑦) ∈ {βˆ…, 𝐴})
4645a1i 11 . . . . . . . . . 10 (𝐴 ∈ 𝑉 β†’ ((π‘₯ = βˆ… ∧ 𝑦 = βˆ…) β†’ (π‘₯ ∩ 𝑦) ∈ {βˆ…, 𝐴}))
47 simpr 486 . . . . . . . . . . . . . 14 ((π‘₯ = 𝐴 ∧ 𝑦 = βˆ…) β†’ 𝑦 = βˆ…)
4847ineq2d 4173 . . . . . . . . . . . . 13 ((π‘₯ = 𝐴 ∧ 𝑦 = βˆ…) β†’ (π‘₯ ∩ 𝑦) = (π‘₯ ∩ βˆ…))
4948, 43eqtrdi 2789 . . . . . . . . . . . 12 ((π‘₯ = 𝐴 ∧ 𝑦 = βˆ…) β†’ (π‘₯ ∩ 𝑦) = βˆ…)
5049, 5eqeltrdi 2842 . . . . . . . . . . 11 ((π‘₯ = 𝐴 ∧ 𝑦 = βˆ…) β†’ (π‘₯ ∩ 𝑦) ∈ {βˆ…, 𝐴})
5150a1i 11 . . . . . . . . . 10 (𝐴 ∈ 𝑉 β†’ ((π‘₯ = 𝐴 ∧ 𝑦 = βˆ…) β†’ (π‘₯ ∩ 𝑦) ∈ {βˆ…, 𝐴}))
52 simpl 484 . . . . . . . . . . . . . 14 ((π‘₯ = βˆ… ∧ 𝑦 = 𝐴) β†’ π‘₯ = βˆ…)
5352ineq1d 4172 . . . . . . . . . . . . 13 ((π‘₯ = βˆ… ∧ 𝑦 = 𝐴) β†’ (π‘₯ ∩ 𝑦) = (βˆ… ∩ 𝑦))
54 0in 4354 . . . . . . . . . . . . 13 (βˆ… ∩ 𝑦) = βˆ…
5553, 54eqtrdi 2789 . . . . . . . . . . . 12 ((π‘₯ = βˆ… ∧ 𝑦 = 𝐴) β†’ (π‘₯ ∩ 𝑦) = βˆ…)
5655, 5eqeltrdi 2842 . . . . . . . . . . 11 ((π‘₯ = βˆ… ∧ 𝑦 = 𝐴) β†’ (π‘₯ ∩ 𝑦) ∈ {βˆ…, 𝐴})
5756a1i 11 . . . . . . . . . 10 (𝐴 ∈ 𝑉 β†’ ((π‘₯ = βˆ… ∧ 𝑦 = 𝐴) β†’ (π‘₯ ∩ 𝑦) ∈ {βˆ…, 𝐴}))
58 ineq12 4168 . . . . . . . . . . . . . 14 ((π‘₯ = 𝐴 ∧ 𝑦 = 𝐴) β†’ (π‘₯ ∩ 𝑦) = (𝐴 ∩ 𝐴))
5958adantl 483 . . . . . . . . . . . . 13 ((𝐴 ∈ 𝑉 ∧ (π‘₯ = 𝐴 ∧ 𝑦 = 𝐴)) β†’ (π‘₯ ∩ 𝑦) = (𝐴 ∩ 𝐴))
60 inidm 4179 . . . . . . . . . . . . 13 (𝐴 ∩ 𝐴) = 𝐴
6159, 60eqtrdi 2789 . . . . . . . . . . . 12 ((𝐴 ∈ 𝑉 ∧ (π‘₯ = 𝐴 ∧ 𝑦 = 𝐴)) β†’ (π‘₯ ∩ 𝑦) = 𝐴)
6218adantr 482 . . . . . . . . . . . 12 ((𝐴 ∈ 𝑉 ∧ (π‘₯ = 𝐴 ∧ 𝑦 = 𝐴)) β†’ 𝐴 ∈ {βˆ…, 𝐴})
6361, 62eqeltrd 2834 . . . . . . . . . . 11 ((𝐴 ∈ 𝑉 ∧ (π‘₯ = 𝐴 ∧ 𝑦 = 𝐴)) β†’ (π‘₯ ∩ 𝑦) ∈ {βˆ…, 𝐴})
6463ex 414 . . . . . . . . . 10 (𝐴 ∈ 𝑉 β†’ ((π‘₯ = 𝐴 ∧ 𝑦 = 𝐴) β†’ (π‘₯ ∩ 𝑦) ∈ {βˆ…, 𝐴}))
6546, 51, 57, 64ccased 1038 . . . . . . . . 9 (𝐴 ∈ 𝑉 β†’ (((π‘₯ = βˆ… ∨ π‘₯ = 𝐴) ∧ (𝑦 = βˆ… ∨ 𝑦 = 𝐴)) β†’ (π‘₯ ∩ 𝑦) ∈ {βˆ…, 𝐴}))
6665expdimp 454 . . . . . . . 8 ((𝐴 ∈ 𝑉 ∧ (π‘₯ = βˆ… ∨ π‘₯ = 𝐴)) β†’ ((𝑦 = βˆ… ∨ 𝑦 = 𝐴) β†’ (π‘₯ ∩ 𝑦) ∈ {βˆ…, 𝐴}))
6740, 66biimtrid 241 . . . . . . 7 ((𝐴 ∈ 𝑉 ∧ (π‘₯ = βˆ… ∨ π‘₯ = 𝐴)) β†’ (𝑦 ∈ {βˆ…, 𝐴} β†’ (π‘₯ ∩ 𝑦) ∈ {βˆ…, 𝐴}))
6867ralrimiv 3139 . . . . . 6 ((𝐴 ∈ 𝑉 ∧ (π‘₯ = βˆ… ∨ π‘₯ = 𝐴)) β†’ βˆ€π‘¦ ∈ {βˆ…, 𝐴} (π‘₯ ∩ 𝑦) ∈ {βˆ…, 𝐴})
6968ex 414 . . . . 5 (𝐴 ∈ 𝑉 β†’ ((π‘₯ = βˆ… ∨ π‘₯ = 𝐴) β†’ βˆ€π‘¦ ∈ {βˆ…, 𝐴} (π‘₯ ∩ 𝑦) ∈ {βˆ…, 𝐴}))
7038, 69biimtrid 241 . . . 4 (𝐴 ∈ 𝑉 β†’ (π‘₯ ∈ {βˆ…, 𝐴} β†’ βˆ€π‘¦ ∈ {βˆ…, 𝐴} (π‘₯ ∩ 𝑦) ∈ {βˆ…, 𝐴}))
7170ralrimiv 3139 . . 3 (𝐴 ∈ 𝑉 β†’ βˆ€π‘₯ ∈ {βˆ…, 𝐴}βˆ€π‘¦ ∈ {βˆ…, 𝐴} (π‘₯ ∩ 𝑦) ∈ {βˆ…, 𝐴})
72 prex 5390 . . . 4 {βˆ…, 𝐴} ∈ V
73 istopg 22260 . . . 4 ({βˆ…, 𝐴} ∈ V β†’ ({βˆ…, 𝐴} ∈ Top ↔ (βˆ€π‘₯(π‘₯ βŠ† {βˆ…, 𝐴} β†’ βˆͺ π‘₯ ∈ {βˆ…, 𝐴}) ∧ βˆ€π‘₯ ∈ {βˆ…, 𝐴}βˆ€π‘¦ ∈ {βˆ…, 𝐴} (π‘₯ ∩ 𝑦) ∈ {βˆ…, 𝐴})))
7472, 73mp1i 13 . . 3 (𝐴 ∈ 𝑉 β†’ ({βˆ…, 𝐴} ∈ Top ↔ (βˆ€π‘₯(π‘₯ βŠ† {βˆ…, 𝐴} β†’ βˆͺ π‘₯ ∈ {βˆ…, 𝐴}) ∧ βˆ€π‘₯ ∈ {βˆ…, 𝐴}βˆ€π‘¦ ∈ {βˆ…, 𝐴} (π‘₯ ∩ 𝑦) ∈ {βˆ…, 𝐴})))
7536, 71, 74mpbir2and 712 . 2 (𝐴 ∈ 𝑉 β†’ {βˆ…, 𝐴} ∈ Top)
7628eqcomd 2739 . 2 (𝐴 ∈ 𝑉 β†’ 𝐴 = βˆͺ {βˆ…, 𝐴})
77 istopon 22277 . 2 ({βˆ…, 𝐴} ∈ (TopOnβ€˜π΄) ↔ ({βˆ…, 𝐴} ∈ Top ∧ 𝐴 = βˆͺ {βˆ…, 𝐴}))
7875, 76, 77sylanbrc 584 1 (𝐴 ∈ 𝑉 β†’ {βˆ…, 𝐴} ∈ (TopOnβ€˜π΄))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846  βˆ€wal 1540   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  Vcvv 3444   βˆͺ cun 3909   ∩ cin 3910   βŠ† wss 3911  βˆ…c0 4283  {csn 4587  {cpr 4589  βˆͺ cuni 4866  β€˜cfv 6497  Topctop 22258  TopOnctopon 22275
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 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-top 22259  df-topon 22276
This theorem is referenced by:  indistop  22368  indisuni  22369  indistpsx  22376  indistpsALT  22379  indistpsALTOLD  22380  indistps2ALT  22381  cnindis  22659  indishmph  23165  indistgp  23467  topdifinf  35866
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