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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 32956 | . 2 ⊢ On ⊆ TopBases | |
2 | indistop 21184 | . . . 4 ⊢ {∅, {{∅}}} ∈ Top | |
3 | topbas 21154 | . . . 4 ⊢ ({∅, {{∅}}} ∈ Top → {∅, {{∅}}} ∈ TopBases) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ {∅, {{∅}}} ∈ TopBases |
5 | snex 5131 | . . . . . 6 ⊢ {{∅}} ∈ V | |
6 | 5 | prid2 4518 | . . . . 5 ⊢ {{∅}} ∈ {∅, {{∅}}} |
7 | snsn0non 6085 | . . . . 5 ⊢ ¬ {{∅}} ∈ On | |
8 | mth8 160 | . . . . 5 ⊢ ({{∅}} ∈ {∅, {{∅}}} → (¬ {{∅}} ∈ On → ¬ ({{∅}} ∈ {∅, {{∅}}} → {{∅}} ∈ On))) | |
9 | 6, 7, 8 | mp2 9 | . . . 4 ⊢ ¬ ({{∅}} ∈ {∅, {{∅}}} → {{∅}} ∈ On) |
10 | onelon 5992 | . . . . 5 ⊢ (({∅, {{∅}}} ∈ On ∧ {{∅}} ∈ {∅, {{∅}}}) → {{∅}} ∈ On) | |
11 | 10 | ex 403 | . . . 4 ⊢ ({∅, {{∅}}} ∈ On → ({{∅}} ∈ {∅, {{∅}}} → {{∅}} ∈ On)) |
12 | 9, 11 | mto 189 | . . 3 ⊢ ¬ {∅, {{∅}}} ∈ On |
13 | 4, 12 | pm3.2i 464 | . 2 ⊢ ({∅, {{∅}}} ∈ TopBases ∧ ¬ {∅, {{∅}}} ∈ On) |
14 | ssnelpss 3946 | . 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 386 ∈ wcel 2164 ⊆ wss 3798 ⊊ wpss 3799 ∅c0 4146 {csn 4399 {cpr 4401 Oncon0 5967 Topctop 21075 TopBasesctb 21127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-ord 5970 df-on 5971 df-iota 6090 df-fun 6129 df-fv 6135 df-top 21076 df-topon 21093 df-bases 21128 |
This theorem is referenced by: (None) |
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