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Mirrors > Home > MPE Home > Th. List > indistpsALT | Structured version Visualization version GIF version |
Description: The indiscrete topology on a set 𝐴 expressed as a topological space. Here we show how to derive the structural version indistps 21762 from the direct component assignment version indistps2 21763. (Contributed by NM, 24-Oct-2012.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
indistpsALT.a | ⊢ 𝐴 ∈ V |
indistpsALT.k | ⊢ 𝐾 = {〈(Base‘ndx), 𝐴〉, 〈(TopSet‘ndx), {∅, 𝐴}〉} |
Ref | Expression |
---|---|
indistpsALT | ⊢ 𝐾 ∈ TopSp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | indistpsALT.a | . 2 ⊢ 𝐴 ∈ V | |
2 | indistopon 21752 | . 2 ⊢ (𝐴 ∈ V → {∅, 𝐴} ∈ (TopOn‘𝐴)) | |
3 | indistpsALT.k | . . . . 5 ⊢ 𝐾 = {〈(Base‘ndx), 𝐴〉, 〈(TopSet‘ndx), {∅, 𝐴}〉} | |
4 | df-tset 16687 | . . . . 5 ⊢ TopSet = Slot 9 | |
5 | 1lt9 11922 | . . . . 5 ⊢ 1 < 9 | |
6 | 9nn 11814 | . . . . 5 ⊢ 9 ∈ ℕ | |
7 | 3, 4, 5, 6 | 2strbas 16706 | . . . 4 ⊢ (𝐴 ∈ V → 𝐴 = (Base‘𝐾)) |
8 | 1, 7 | ax-mp 5 | . . 3 ⊢ 𝐴 = (Base‘𝐾) |
9 | prex 5299 | . . . 4 ⊢ {∅, 𝐴} ∈ V | |
10 | 3, 4, 5, 6 | 2strop 16707 | . . . 4 ⊢ ({∅, 𝐴} ∈ V → {∅, 𝐴} = (TopSet‘𝐾)) |
11 | 9, 10 | ax-mp 5 | . . 3 ⊢ {∅, 𝐴} = (TopSet‘𝐾) |
12 | 8, 11 | tsettps 21692 | . 2 ⊢ ({∅, 𝐴} ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp) |
13 | 1, 2, 12 | mp2b 10 | 1 ⊢ 𝐾 ∈ TopSp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2114 Vcvv 3398 ∅c0 4211 {cpr 4518 〈cop 4522 ‘cfv 6339 9c9 11778 ndxcnx 16583 Basecbs 16586 TopSetcts 16674 TopOnctopon 21661 TopSpctps 21683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-uz 12325 df-fz 12982 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-tset 16687 df-rest 16799 df-topn 16800 df-top 21645 df-topon 21662 df-topsp 21684 |
This theorem is referenced by: (None) |
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