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Theorem dissneq 35073
 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 4535 . . . . . 6 (𝑧 = 𝑥 → {𝑧} = {𝑥})
32eqeq2d 2769 . . . . 5 (𝑧 = 𝑥 → (𝑢 = {𝑧} ↔ 𝑢 = {𝑥}))
43cbvrexvw 3362 . . . 4 (∃𝑧𝐴 𝑢 = {𝑧} ↔ ∃𝑥𝐴 𝑢 = {𝑥})
54abbii 2823 . . 3 {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}} = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
61, 5eqtr4i 2784 . 2 𝐶 = {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}}
76dissneqlem 35072 1 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → 𝐵 = 𝒫 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2735  ∃wrex 3071   ⊆ wss 3860  𝒫 cpw 4497  {csn 4525  ‘cfv 6340  TopOnctopon 21624 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fv 6348  df-topgen 16789  df-top 21608  df-topon 21625 This theorem is referenced by:  topdifinffinlem  35079
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