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 31598 | . 2 ⊢ On ⊆ TopBases | |
2 | indistop 20616 | . . . 4 ⊢ {∅, {{∅}}} ∈ Top | |
3 | topbas 20587 | . . . 4 ⊢ ({∅, {{∅}}} ∈ Top → {∅, {{∅}}} ∈ TopBases) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ {∅, {{∅}}} ∈ TopBases |
5 | snex 4835 | . . . . . 6 ⊢ {{∅}} ∈ V | |
6 | 5 | prid2 4242 | . . . . 5 ⊢ {{∅}} ∈ {∅, {{∅}}} |
7 | snsn0non 5763 | . . . . 5 ⊢ ¬ {{∅}} ∈ On | |
8 | mth8 157 | . . . . 5 ⊢ ({{∅}} ∈ {∅, {{∅}}} → (¬ {{∅}} ∈ On → ¬ ({{∅}} ∈ {∅, {{∅}}} → {{∅}} ∈ On))) | |
9 | 6, 7, 8 | mp2 9 | . . . 4 ⊢ ¬ ({{∅}} ∈ {∅, {{∅}}} → {{∅}} ∈ On) |
10 | onelon 5665 | . . . . 5 ⊢ (({∅, {{∅}}} ∈ On ∧ {{∅}} ∈ {∅, {{∅}}}) → {{∅}} ∈ On) | |
11 | 10 | ex 449 | . . . 4 ⊢ ({∅, {{∅}}} ∈ On → ({{∅}} ∈ {∅, {{∅}}} → {{∅}} ∈ On)) |
12 | 9, 11 | mto 187 | . . 3 ⊢ ¬ {∅, {{∅}}} ∈ On |
13 | 4, 12 | pm3.2i 470 | . 2 ⊢ ({∅, {{∅}}} ∈ TopBases ∧ ¬ {∅, {{∅}}} ∈ On) |
14 | ssnelpss 3680 | . 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 383 ∈ wcel 1977 ⊆ wss 3540 ⊊ wpss 3541 ∅c0 3874 {csn 4125 {cpr 4127 Oncon0 5640 Topctop 20517 TopBasesctb 20520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-ord 5643 df-on 5644 df-iota 5768 df-fun 5806 df-fv 5812 df-top 20521 df-bases 20522 df-topon 20523 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |