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Theorem dissneq 32364
 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 4135 . . . . . 6 (𝑧 = 𝑥 → {𝑧} = {𝑥})
32eqeq2d 2620 . . . . 5 (𝑧 = 𝑥 → (𝑢 = {𝑧} ↔ 𝑢 = {𝑥}))
43cbvrexv 3148 . . . 4 (∃𝑧𝐴 𝑢 = {𝑧} ↔ ∃𝑥𝐴 𝑢 = {𝑥})
54abbii 2726 . . 3 {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}} = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
61, 5eqtr4i 2635 . 2 𝐶 = {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}}
76dissneqlem 32363 1 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → 𝐵 = 𝒫 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∃wrex 2897   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  ‘cfv 5804  TopOnctopon 20518 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-topgen 15927  df-top 20521  df-topon 20523 This theorem is referenced by:  topdifinffinlem  32371
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