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Theorem indistopon 22511
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 4836 . . . . 5 (π‘₯ βŠ† {βˆ…, 𝐴} ↔ ((π‘₯ = βˆ… ∨ π‘₯ = {βˆ…}) ∨ (π‘₯ = {𝐴} ∨ π‘₯ = {βˆ…, 𝐴})))
2 unieq 4919 . . . . . . . . 9 (π‘₯ = βˆ… β†’ βˆͺ π‘₯ = βˆͺ βˆ…)
3 uni0 4939 . . . . . . . . . 10 βˆͺ βˆ… = βˆ…
4 0ex 5307 . . . . . . . . . . 11 βˆ… ∈ V
54prid1 4766 . . . . . . . . . 10 βˆ… ∈ {βˆ…, 𝐴}
63, 5eqeltri 2829 . . . . . . . . 9 βˆͺ βˆ… ∈ {βˆ…, 𝐴}
72, 6eqeltrdi 2841 . . . . . . . 8 (π‘₯ = βˆ… β†’ βˆͺ π‘₯ ∈ {βˆ…, 𝐴})
87a1i 11 . . . . . . 7 (𝐴 ∈ 𝑉 β†’ (π‘₯ = βˆ… β†’ βˆͺ π‘₯ ∈ {βˆ…, 𝐴}))
9 unieq 4919 . . . . . . . . 9 (π‘₯ = {βˆ…} β†’ βˆͺ π‘₯ = βˆͺ {βˆ…})
104unisn 4930 . . . . . . . . . 10 βˆͺ {βˆ…} = βˆ…
1110, 5eqeltri 2829 . . . . . . . . 9 βˆͺ {βˆ…} ∈ {βˆ…, 𝐴}
129, 11eqeltrdi 2841 . . . . . . . 8 (π‘₯ = {βˆ…} β†’ βˆͺ π‘₯ ∈ {βˆ…, 𝐴})
1312a1i 11 . . . . . . 7 (𝐴 ∈ 𝑉 β†’ (π‘₯ = {βˆ…} β†’ βˆͺ π‘₯ ∈ {βˆ…, 𝐴}))
148, 13jaod 857 . . . . . 6 (𝐴 ∈ 𝑉 β†’ ((π‘₯ = βˆ… ∨ π‘₯ = {βˆ…}) β†’ βˆͺ π‘₯ ∈ {βˆ…, 𝐴}))
15 unieq 4919 . . . . . . . . . 10 (π‘₯ = {𝐴} β†’ βˆͺ π‘₯ = βˆͺ {𝐴})
16 unisng 4929 . . . . . . . . . 10 (𝐴 ∈ 𝑉 β†’ βˆͺ {𝐴} = 𝐴)
1715, 16sylan9eqr 2794 . . . . . . . . 9 ((𝐴 ∈ 𝑉 ∧ π‘₯ = {𝐴}) β†’ βˆͺ π‘₯ = 𝐴)
18 prid2g 4765 . . . . . . . . . 10 (𝐴 ∈ 𝑉 β†’ 𝐴 ∈ {βˆ…, 𝐴})
1918adantr 481 . . . . . . . . 9 ((𝐴 ∈ 𝑉 ∧ π‘₯ = {𝐴}) β†’ 𝐴 ∈ {βˆ…, 𝐴})
2017, 19eqeltrd 2833 . . . . . . . 8 ((𝐴 ∈ 𝑉 ∧ π‘₯ = {𝐴}) β†’ βˆͺ π‘₯ ∈ {βˆ…, 𝐴})
2120ex 413 . . . . . . 7 (𝐴 ∈ 𝑉 β†’ (π‘₯ = {𝐴} β†’ βˆͺ π‘₯ ∈ {βˆ…, 𝐴}))
22 unieq 4919 . . . . . . . . . 10 (π‘₯ = {βˆ…, 𝐴} β†’ βˆͺ π‘₯ = βˆͺ {βˆ…, 𝐴})
23 uniprg 4925 . . . . . . . . . . . 12 ((βˆ… ∈ V ∧ 𝐴 ∈ 𝑉) β†’ βˆͺ {βˆ…, 𝐴} = (βˆ… βˆͺ 𝐴))
244, 23mpan 688 . . . . . . . . . . 11 (𝐴 ∈ 𝑉 β†’ βˆͺ {βˆ…, 𝐴} = (βˆ… βˆͺ 𝐴))
25 uncom 4153 . . . . . . . . . . . 12 (βˆ… βˆͺ 𝐴) = (𝐴 βˆͺ βˆ…)
26 un0 4390 . . . . . . . . . . . 12 (𝐴 βˆͺ βˆ…) = 𝐴
2725, 26eqtri 2760 . . . . . . . . . . 11 (βˆ… βˆͺ 𝐴) = 𝐴
2824, 27eqtrdi 2788 . . . . . . . . . 10 (𝐴 ∈ 𝑉 β†’ βˆͺ {βˆ…, 𝐴} = 𝐴)
2922, 28sylan9eqr 2794 . . . . . . . . 9 ((𝐴 ∈ 𝑉 ∧ π‘₯ = {βˆ…, 𝐴}) β†’ βˆͺ π‘₯ = 𝐴)
3018adantr 481 . . . . . . . . 9 ((𝐴 ∈ 𝑉 ∧ π‘₯ = {βˆ…, 𝐴}) β†’ 𝐴 ∈ {βˆ…, 𝐴})
3129, 30eqeltrd 2833 . . . . . . . 8 ((𝐴 ∈ 𝑉 ∧ π‘₯ = {βˆ…, 𝐴}) β†’ βˆͺ π‘₯ ∈ {βˆ…, 𝐴})
3231ex 413 . . . . . . 7 (𝐴 ∈ 𝑉 β†’ (π‘₯ = {βˆ…, 𝐴} β†’ βˆͺ π‘₯ ∈ {βˆ…, 𝐴}))
3321, 32jaod 857 . . . . . 6 (𝐴 ∈ 𝑉 β†’ ((π‘₯ = {𝐴} ∨ π‘₯ = {βˆ…, 𝐴}) β†’ βˆͺ π‘₯ ∈ {βˆ…, 𝐴}))
3414, 33jaod 857 . . . . 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 1930 . . 3 (𝐴 ∈ 𝑉 β†’ βˆ€π‘₯(π‘₯ βŠ† {βˆ…, 𝐴} β†’ βˆͺ π‘₯ ∈ {βˆ…, 𝐴}))
37 vex 3478 . . . . . 6 π‘₯ ∈ V
3837elpr 4651 . . . . 5 (π‘₯ ∈ {βˆ…, 𝐴} ↔ (π‘₯ = βˆ… ∨ π‘₯ = 𝐴))
39 vex 3478 . . . . . . . . 9 𝑦 ∈ V
4039elpr 4651 . . . . . . . 8 (𝑦 ∈ {βˆ…, 𝐴} ↔ (𝑦 = βˆ… ∨ 𝑦 = 𝐴))
41 simpr 485 . . . . . . . . . . . . . 14 ((π‘₯ = βˆ… ∧ 𝑦 = βˆ…) β†’ 𝑦 = βˆ…)
4241ineq2d 4212 . . . . . . . . . . . . 13 ((π‘₯ = βˆ… ∧ 𝑦 = βˆ…) β†’ (π‘₯ ∩ 𝑦) = (π‘₯ ∩ βˆ…))
43 in0 4391 . . . . . . . . . . . . 13 (π‘₯ ∩ βˆ…) = βˆ…
4442, 43eqtrdi 2788 . . . . . . . . . . . 12 ((π‘₯ = βˆ… ∧ 𝑦 = βˆ…) β†’ (π‘₯ ∩ 𝑦) = βˆ…)
4544, 5eqeltrdi 2841 . . . . . . . . . . 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 485 . . . . . . . . . . . . . 14 ((π‘₯ = 𝐴 ∧ 𝑦 = βˆ…) β†’ 𝑦 = βˆ…)
4847ineq2d 4212 . . . . . . . . . . . . 13 ((π‘₯ = 𝐴 ∧ 𝑦 = βˆ…) β†’ (π‘₯ ∩ 𝑦) = (π‘₯ ∩ βˆ…))
4948, 43eqtrdi 2788 . . . . . . . . . . . 12 ((π‘₯ = 𝐴 ∧ 𝑦 = βˆ…) β†’ (π‘₯ ∩ 𝑦) = βˆ…)
5049, 5eqeltrdi 2841 . . . . . . . . . . 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 483 . . . . . . . . . . . . . 14 ((π‘₯ = βˆ… ∧ 𝑦 = 𝐴) β†’ π‘₯ = βˆ…)
5352ineq1d 4211 . . . . . . . . . . . . 13 ((π‘₯ = βˆ… ∧ 𝑦 = 𝐴) β†’ (π‘₯ ∩ 𝑦) = (βˆ… ∩ 𝑦))
54 0in 4393 . . . . . . . . . . . . 13 (βˆ… ∩ 𝑦) = βˆ…
5553, 54eqtrdi 2788 . . . . . . . . . . . 12 ((π‘₯ = βˆ… ∧ 𝑦 = 𝐴) β†’ (π‘₯ ∩ 𝑦) = βˆ…)
5655, 5eqeltrdi 2841 . . . . . . . . . . 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 4207 . . . . . . . . . . . . . 14 ((π‘₯ = 𝐴 ∧ 𝑦 = 𝐴) β†’ (π‘₯ ∩ 𝑦) = (𝐴 ∩ 𝐴))
5958adantl 482 . . . . . . . . . . . . 13 ((𝐴 ∈ 𝑉 ∧ (π‘₯ = 𝐴 ∧ 𝑦 = 𝐴)) β†’ (π‘₯ ∩ 𝑦) = (𝐴 ∩ 𝐴))
60 inidm 4218 . . . . . . . . . . . . 13 (𝐴 ∩ 𝐴) = 𝐴
6159, 60eqtrdi 2788 . . . . . . . . . . . 12 ((𝐴 ∈ 𝑉 ∧ (π‘₯ = 𝐴 ∧ 𝑦 = 𝐴)) β†’ (π‘₯ ∩ 𝑦) = 𝐴)
6218adantr 481 . . . . . . . . . . . 12 ((𝐴 ∈ 𝑉 ∧ (π‘₯ = 𝐴 ∧ 𝑦 = 𝐴)) β†’ 𝐴 ∈ {βˆ…, 𝐴})
6361, 62eqeltrd 2833 . . . . . . . . . . 11 ((𝐴 ∈ 𝑉 ∧ (π‘₯ = 𝐴 ∧ 𝑦 = 𝐴)) β†’ (π‘₯ ∩ 𝑦) ∈ {βˆ…, 𝐴})
6463ex 413 . . . . . . . . . 10 (𝐴 ∈ 𝑉 β†’ ((π‘₯ = 𝐴 ∧ 𝑦 = 𝐴) β†’ (π‘₯ ∩ 𝑦) ∈ {βˆ…, 𝐴}))
6546, 51, 57, 64ccased 1037 . . . . . . . . 9 (𝐴 ∈ 𝑉 β†’ (((π‘₯ = βˆ… ∨ π‘₯ = 𝐴) ∧ (𝑦 = βˆ… ∨ 𝑦 = 𝐴)) β†’ (π‘₯ ∩ 𝑦) ∈ {βˆ…, 𝐴}))
6665expdimp 453 . . . . . . . 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 3145 . . . . . 6 ((𝐴 ∈ 𝑉 ∧ (π‘₯ = βˆ… ∨ π‘₯ = 𝐴)) β†’ βˆ€π‘¦ ∈ {βˆ…, 𝐴} (π‘₯ ∩ 𝑦) ∈ {βˆ…, 𝐴})
6968ex 413 . . . . 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 3145 . . 3 (𝐴 ∈ 𝑉 β†’ βˆ€π‘₯ ∈ {βˆ…, 𝐴}βˆ€π‘¦ ∈ {βˆ…, 𝐴} (π‘₯ ∩ 𝑦) ∈ {βˆ…, 𝐴})
72 prex 5432 . . . 4 {βˆ…, 𝐴} ∈ V
73 istopg 22404 . . . 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 711 . 2 (𝐴 ∈ 𝑉 β†’ {βˆ…, 𝐴} ∈ Top)
7628eqcomd 2738 . 2 (𝐴 ∈ 𝑉 β†’ 𝐴 = βˆͺ {βˆ…, 𝐴})
77 istopon 22421 . 2 ({βˆ…, 𝐴} ∈ (TopOnβ€˜π΄) ↔ ({βˆ…, 𝐴} ∈ Top ∧ 𝐴 = βˆͺ {βˆ…, 𝐴}))
7875, 76, 77sylanbrc 583 1 (𝐴 ∈ 𝑉 β†’ {βˆ…, 𝐴} ∈ (TopOnβ€˜π΄))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845  βˆ€wal 1539   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  Vcvv 3474   βˆͺ cun 3946   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  {csn 4628  {cpr 4630  βˆͺ cuni 4908  β€˜cfv 6543  Topctop 22402  TopOnctopon 22419
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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-top 22403  df-topon 22420
This theorem is referenced by:  indistop  22512  indisuni  22513  indistpsx  22520  indistpsALT  22523  indistpsALTOLD  22524  indistps2ALT  22525  cnindis  22803  indishmph  23309  indistgp  23611  topdifinf  36322
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