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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 32428 | . 2 ⊢ On ⊆ TopBases | |
2 | indistop 20806 | . . . 4 ⊢ {∅, {{∅}}} ∈ Top | |
3 | topbas 20776 | . . . 4 ⊢ ({∅, {{∅}}} ∈ Top → {∅, {{∅}}} ∈ TopBases) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ {∅, {{∅}}} ∈ TopBases |
5 | snex 4908 | . . . . . 6 ⊢ {{∅}} ∈ V | |
6 | 5 | prid2 4298 | . . . . 5 ⊢ {{∅}} ∈ {∅, {{∅}}} |
7 | snsn0non 5846 | . . . . 5 ⊢ ¬ {{∅}} ∈ On | |
8 | mth8 158 | . . . . 5 ⊢ ({{∅}} ∈ {∅, {{∅}}} → (¬ {{∅}} ∈ On → ¬ ({{∅}} ∈ {∅, {{∅}}} → {{∅}} ∈ On))) | |
9 | 6, 7, 8 | mp2 9 | . . . 4 ⊢ ¬ ({{∅}} ∈ {∅, {{∅}}} → {{∅}} ∈ On) |
10 | onelon 5748 | . . . . 5 ⊢ (({∅, {{∅}}} ∈ On ∧ {{∅}} ∈ {∅, {{∅}}}) → {{∅}} ∈ On) | |
11 | 10 | ex 450 | . . . 4 ⊢ ({∅, {{∅}}} ∈ On → ({{∅}} ∈ {∅, {{∅}}} → {{∅}} ∈ On)) |
12 | 9, 11 | mto 188 | . . 3 ⊢ ¬ {∅, {{∅}}} ∈ On |
13 | 4, 12 | pm3.2i 471 | . 2 ⊢ ({∅, {{∅}}} ∈ TopBases ∧ ¬ {∅, {{∅}}} ∈ On) |
14 | ssnelpss 3718 | . 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 384 ∈ wcel 1990 ⊆ wss 3574 ⊊ wpss 3575 ∅c0 3915 {csn 4177 {cpr 4179 Oncon0 5723 Topctop 20698 TopBasesctb 20749 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-ord 5726 df-on 5727 df-iota 5851 df-fun 5890 df-fv 5896 df-top 20699 df-topon 20716 df-bases 20750 |
This theorem is referenced by: (None) |
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