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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 33772 | . 2 ⊢ On ⊆ TopBases | |
2 | indistop 21604 | . . . 4 ⊢ {∅, {{∅}}} ∈ Top | |
3 | topbas 21574 | . . . 4 ⊢ ({∅, {{∅}}} ∈ Top → {∅, {{∅}}} ∈ TopBases) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ {∅, {{∅}}} ∈ TopBases |
5 | snex 5324 | . . . . . 6 ⊢ {{∅}} ∈ V | |
6 | 5 | prid2 4693 | . . . . 5 ⊢ {{∅}} ∈ {∅, {{∅}}} |
7 | snsn0non 6304 | . . . . 5 ⊢ ¬ {{∅}} ∈ On | |
8 | jcn 164 | . . . . 5 ⊢ ({{∅}} ∈ {∅, {{∅}}} → (¬ {{∅}} ∈ On → ¬ ({{∅}} ∈ {∅, {{∅}}} → {{∅}} ∈ On))) | |
9 | 6, 7, 8 | mp2 9 | . . . 4 ⊢ ¬ ({{∅}} ∈ {∅, {{∅}}} → {{∅}} ∈ On) |
10 | onelon 6211 | . . . . 5 ⊢ (({∅, {{∅}}} ∈ On ∧ {{∅}} ∈ {∅, {{∅}}}) → {{∅}} ∈ On) | |
11 | 10 | ex 415 | . . . 4 ⊢ ({∅, {{∅}}} ∈ On → ({{∅}} ∈ {∅, {{∅}}} → {{∅}} ∈ On)) |
12 | 9, 11 | mto 199 | . . 3 ⊢ ¬ {∅, {{∅}}} ∈ On |
13 | 4, 12 | pm3.2i 473 | . 2 ⊢ ({∅, {{∅}}} ∈ TopBases ∧ ¬ {∅, {{∅}}} ∈ On) |
14 | ssnelpss 4088 | . 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 398 ∈ wcel 2110 ⊆ wss 3936 ⊊ wpss 3937 ∅c0 4291 {csn 4561 {cpr 4563 Oncon0 6186 Topctop 21495 TopBasesctb 21547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-ord 6189 df-on 6190 df-iota 6309 df-fun 6352 df-fv 6358 df-top 21496 df-topon 21513 df-bases 21548 |
This theorem is referenced by: (None) |
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