Theorem topnex 22890

Description: The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 7738; an alternate proof uses indiscrete topologies (see indistop 22896) and the analogue of pwnex 7738 with pairs {∅, 𝑥} instead of power sets 𝒫 𝑥 (that analogue is also a consequence of abnex 7736). (Contributed by BJ, 2-May-2021.)

Assertion

Ref	Expression
topnex	⊢ Top ∉ V

Proof of Theorem topnex

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwnex 7738	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∉ V
2	1	neli 3032	. . 3 ⊢ ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V
3		distop 22889	. . . . . . . 8 ⊢ (𝑥 ∈ V → 𝒫 𝑥 ∈ Top)
4	3	elv 3455	. . . . . . 7 ⊢ 𝒫 𝑥 ∈ Top
5		eleq1 2817	. . . . . . 7 ⊢ (𝑦 = 𝒫 𝑥 → (𝑦 ∈ Top ↔ 𝒫 𝑥 ∈ Top))
6	4, 5	mpbiri 258	. . . . . 6 ⊢ (𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top)
7	6	exlimiv 1930	. . . . 5 ⊢ (∃𝑥 𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top)
8	7	abssi 4036	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top
9		ssexg 5281	. . . 4 ⊢ (({𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top ∧ Top ∈ V) → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
10	8, 9	mpan 690	. . 3 ⊢ (Top ∈ V → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V)
11	2, 10	mto 197	. 2 ⊢ ¬ Top ∈ V
12	11	nelir 3033	1 ⊢ Top ∉ V

Syntax hints:   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2708   ∉ wnel 3030  Vcvv 3450   ⊆ wss 3917  𝒫 cpw 4566  Topctop 22787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-ext 2702  ax-sep 5254  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nel 3031  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-un 3922  df-in 3924  df-ss 3934  df-pw 4568  df-sn 4593  df-pr 4595  df-uni 4875  df-iun 4960  df-top 22788

This theorem is referenced by: (None)