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Mirrors > Home > MPE Home > Th. List > topnex | Structured version Visualization version GIF version |
Description: The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 7481; an alternate proof uses indiscrete topologies (see indistop 21610) and the analogue of pwnex 7481 with pairs {∅, 𝑥} instead of power sets 𝒫 𝑥 (that analogue is also a consequence of abnex 7479). (Contributed by BJ, 2-May-2021.) |
Ref | Expression |
---|---|
topnex | ⊢ Top ∉ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwnex 7481 | . . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∉ V | |
2 | 1 | neli 3125 | . . 3 ⊢ ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V |
3 | distop 21603 | . . . . . . . 8 ⊢ (𝑥 ∈ V → 𝒫 𝑥 ∈ Top) | |
4 | 3 | elv 3499 | . . . . . . 7 ⊢ 𝒫 𝑥 ∈ Top |
5 | eleq1 2900 | . . . . . . 7 ⊢ (𝑦 = 𝒫 𝑥 → (𝑦 ∈ Top ↔ 𝒫 𝑥 ∈ Top)) | |
6 | 4, 5 | mpbiri 260 | . . . . . 6 ⊢ (𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top) |
7 | 6 | exlimiv 1931 | . . . . 5 ⊢ (∃𝑥 𝑦 = 𝒫 𝑥 → 𝑦 ∈ Top) |
8 | 7 | abssi 4046 | . . . 4 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top |
9 | ssexg 5227 | . . . 4 ⊢ (({𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ⊆ Top ∧ Top ∈ V) → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V) | |
10 | 8, 9 | mpan 688 | . . 3 ⊢ (Top ∈ V → {𝑦 ∣ ∃𝑥 𝑦 = 𝒫 𝑥} ∈ V) |
11 | 2, 10 | mto 199 | . 2 ⊢ ¬ Top ∈ V |
12 | 11 | nelir 3126 | 1 ⊢ Top ∉ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∃wex 1780 ∈ wcel 2114 {cab 2799 ∉ wnel 3123 Vcvv 3494 ⊆ wss 3936 𝒫 cpw 4539 Topctop 21501 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-pw 4541 df-sn 4568 df-pr 4570 df-uni 4839 df-iun 4921 df-top 21502 |
This theorem is referenced by: (None) |
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