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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 36430 | . 2 ⊢ On ⊆ TopBases | |
| 2 | indistop 23009 | . . . 4 ⊢ {∅, {{∅}}} ∈ Top | |
| 3 | topbas 22979 | . . . 4 ⊢ ({∅, {{∅}}} ∈ Top → {∅, {{∅}}} ∈ TopBases) | |
| 4 | 2, 3 | ax-mp 5 | . . 3 ⊢ {∅, {{∅}}} ∈ TopBases |
| 5 | snex 5436 | . . . . . 6 ⊢ {{∅}} ∈ V | |
| 6 | 5 | prid2 4763 | . . . . 5 ⊢ {{∅}} ∈ {∅, {{∅}}} |
| 7 | snsn0non 6509 | . . . . 5 ⊢ ¬ {{∅}} ∈ On | |
| 8 | jcn 162 | . . . . 5 ⊢ ({{∅}} ∈ {∅, {{∅}}} → (¬ {{∅}} ∈ On → ¬ ({{∅}} ∈ {∅, {{∅}}} → {{∅}} ∈ On))) | |
| 9 | 6, 7, 8 | mp2 9 | . . . 4 ⊢ ¬ ({{∅}} ∈ {∅, {{∅}}} → {{∅}} ∈ On) |
| 10 | onelon 6409 | . . . . 5 ⊢ (({∅, {{∅}}} ∈ On ∧ {{∅}} ∈ {∅, {{∅}}}) → {{∅}} ∈ On) | |
| 11 | 10 | ex 412 | . . . 4 ⊢ ({∅, {{∅}}} ∈ On → ({{∅}} ∈ {∅, {{∅}}} → {{∅}} ∈ On)) |
| 12 | 9, 11 | mto 197 | . . 3 ⊢ ¬ {∅, {{∅}}} ∈ On |
| 13 | 4, 12 | pm3.2i 470 | . 2 ⊢ ({∅, {{∅}}} ∈ TopBases ∧ ¬ {∅, {{∅}}} ∈ On) |
| 14 | ssnelpss 4114 | . 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 395 ∈ wcel 2108 ⊆ wss 3951 ⊊ wpss 3952 ∅c0 4333 {csn 4626 {cpr 4628 Oncon0 6384 Topctop 22899 TopBasesctb 22952 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-ord 6387 df-on 6388 df-iota 6514 df-fun 6563 df-fv 6569 df-top 22900 df-topon 22917 df-bases 22953 |
| This theorem is referenced by: (None) |
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