![]() |
Mathbox for Chen-Pang He |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > onpsstopbas | Structured version Visualization version GIF version |
Description: The class of ordinal numbers is a proper subclass of the class of topological bases. (Contributed by Chen-Pang He, 9-Oct-2015.) |
Ref | Expression |
---|---|
onpsstopbas | ⊢ On ⊊ TopBases |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onsstopbas 33890 | . 2 ⊢ On ⊆ TopBases | |
2 | indistop 21607 | . . . 4 ⊢ {∅, {{∅}}} ∈ Top | |
3 | topbas 21577 | . . . 4 ⊢ ({∅, {{∅}}} ∈ Top → {∅, {{∅}}} ∈ TopBases) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ {∅, {{∅}}} ∈ TopBases |
5 | snex 5297 | . . . . . 6 ⊢ {{∅}} ∈ V | |
6 | 5 | prid2 4659 | . . . . 5 ⊢ {{∅}} ∈ {∅, {{∅}}} |
7 | snsn0non 6277 | . . . . 5 ⊢ ¬ {{∅}} ∈ On | |
8 | jcn 165 | . . . . 5 ⊢ ({{∅}} ∈ {∅, {{∅}}} → (¬ {{∅}} ∈ On → ¬ ({{∅}} ∈ {∅, {{∅}}} → {{∅}} ∈ On))) | |
9 | 6, 7, 8 | mp2 9 | . . . 4 ⊢ ¬ ({{∅}} ∈ {∅, {{∅}}} → {{∅}} ∈ On) |
10 | onelon 6184 | . . . . 5 ⊢ (({∅, {{∅}}} ∈ On ∧ {{∅}} ∈ {∅, {{∅}}}) → {{∅}} ∈ On) | |
11 | 10 | ex 416 | . . . 4 ⊢ ({∅, {{∅}}} ∈ On → ({{∅}} ∈ {∅, {{∅}}} → {{∅}} ∈ On)) |
12 | 9, 11 | mto 200 | . . 3 ⊢ ¬ {∅, {{∅}}} ∈ On |
13 | 4, 12 | pm3.2i 474 | . 2 ⊢ ({∅, {{∅}}} ∈ TopBases ∧ ¬ {∅, {{∅}}} ∈ On) |
14 | ssnelpss 4039 | . 2 ⊢ (On ⊆ TopBases → (({∅, {{∅}}} ∈ TopBases ∧ ¬ {∅, {{∅}}} ∈ On) → On ⊊ TopBases)) | |
15 | 1, 13, 14 | mp2 9 | 1 ⊢ On ⊊ TopBases |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∈ wcel 2111 ⊆ wss 3881 ⊊ wpss 3882 ∅c0 4243 {csn 4525 {cpr 4527 Oncon0 6159 Topctop 21498 TopBasesctb 21550 |
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 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-ord 6162 df-on 6163 df-iota 6283 df-fun 6326 df-fv 6332 df-top 21499 df-topon 21516 df-bases 21551 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |