Theorem dissneq 38532

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 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}

Assertion

Ref	Expression
dissneq	⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (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.

Step	Hyp	Ref	Expression
1		dissneq.c	. . 3 ⊢ 𝐶 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}
2		sneq 4594	. . . . . 6 ⊢ (𝑧 = 𝑥 → {𝑧} = {𝑥})
3	2	eqeq2d 2774	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝑢 = {𝑧} ↔ 𝑢 = {𝑥}))
4	3	cbvrexvw 3245	. . . 4 ⊢ (∃𝑧 ∈ 𝐴 𝑢 = {𝑧} ↔ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥})
5	4	abbii 2830	. . 3 ⊢ {𝑢 ∣ ∃𝑧 ∈ 𝐴 𝑢 = {𝑧}} = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}
6	1, 5	eqtr4i 2789	. 2 ⊢ 𝐶 = {𝑢 ∣ ∃𝑧 ∈ 𝐴 𝑢 = {𝑧}}
7	6	dissneqlem 38531	1 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → 𝐵 = 𝒫 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  ∃wrex 3090   ⊆ wss 3900  𝒫 cpw 4560  {csn 4588  ‘cfv 6566  TopOnctopon 23283

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fv 6574  df-topgen 17664  df-top 23267  df-topon 23284

This theorem is used by:  topdifinffinlem  38538