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Theorem dissneq 32855
Description: Any topology that contains every single-point set is the discrete topology. (Contributed by ML, 16-Jul-2020.)
Hypothesis
Ref Expression
dissneq.c 𝐶 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
Assertion
Ref Expression
dissneq ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → 𝐵 = 𝒫 𝐴)
Distinct variable group:   𝑢,𝐴,𝑥
Allowed substitution hints:   𝐵(𝑥,𝑢)   𝐶(𝑥,𝑢)

Proof of Theorem dissneq
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dissneq.c . . 3 𝐶 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
2 sneq 4163 . . . . . 6 (𝑧 = 𝑥 → {𝑧} = {𝑥})
32eqeq2d 2631 . . . . 5 (𝑧 = 𝑥 → (𝑢 = {𝑧} ↔ 𝑢 = {𝑥}))
43cbvrexv 3163 . . . 4 (∃𝑧𝐴 𝑢 = {𝑧} ↔ ∃𝑥𝐴 𝑢 = {𝑥})
54abbii 2736 . . 3 {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}} = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
61, 5eqtr4i 2646 . 2 𝐶 = {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}}
76dissneqlem 32854 1 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → 𝐵 = 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {cab 2607  wrex 2908  wss 3559  𝒫 cpw 4135  {csn 4153  cfv 5852  TopOnctopon 20647
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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
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 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  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-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fv 5860  df-topgen 16036  df-top 20631  df-topon 20648
This theorem is referenced by:  topdifinffinlem  32862
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