Theorem dissneq 37375

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 4581	. . . . . 6 ⊢ (𝑧 = 𝑥 → {𝑧} = {𝑥})
3	2	eqeq2d 2742	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝑢 = {𝑧} ↔ 𝑢 = {𝑥}))
4	3	cbvrexvw 3211	. . . 4 ⊢ (∃𝑧 ∈ 𝐴 𝑢 = {𝑧} ↔ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥})
5	4	abbii 2798	. . 3 ⊢ {𝑢 ∣ ∃𝑧 ∈ 𝐴 𝑢 = {𝑧}} = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}
6	1, 5	eqtr4i 2757	. 2 ⊢ 𝐶 = {𝑢 ∣ ∃𝑧 ∈ 𝐴 𝑢 = {𝑧}}
7	6	dissneqlem 37374	1 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → 𝐵 = 𝒫 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {cab 2709  ∃wrex 3056   ⊆ wss 3897  𝒫 cpw 4545  {csn 4571  ‘cfv 6476  TopOnctopon 22820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fv 6484  df-topgen 17342  df-top 22804  df-topon 22821

This theorem is referenced by:  topdifinffinlem  37381