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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 22834 from the direct component assignment version indistps2 22835. (Contributed by NM, 24-Oct-2012.) (Revised by AV, 31-Oct-2024.) (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 22824 | . 2 ⊢ (𝐴 ∈ V → {∅, 𝐴} ∈ (TopOn‘𝐴)) | |
3 | indistpsALT.k | . . . . 5 ⊢ 𝐾 = {〈(Base‘ndx), 𝐴〉, 〈(TopSet‘ndx), {∅, 𝐴}〉} | |
4 | basendxlttsetndx 17307 | . . . . 5 ⊢ (Base‘ndx) < (TopSet‘ndx) | |
5 | tsetndxnn 17306 | . . . . 5 ⊢ (TopSet‘ndx) ∈ ℕ | |
6 | 3, 4, 5 | 2strbas1 17178 | . . . 4 ⊢ (𝐴 ∈ V → 𝐴 = (Base‘𝐾)) |
7 | 1, 6 | ax-mp 5 | . . 3 ⊢ 𝐴 = (Base‘𝐾) |
8 | prex 5432 | . . . 4 ⊢ {∅, 𝐴} ∈ V | |
9 | tsetid 17305 | . . . . 5 ⊢ TopSet = Slot (TopSet‘ndx) | |
10 | 3, 4, 5, 9 | 2strop1 17179 | . . . 4 ⊢ ({∅, 𝐴} ∈ V → {∅, 𝐴} = (TopSet‘𝐾)) |
11 | 8, 10 | ax-mp 5 | . . 3 ⊢ {∅, 𝐴} = (TopSet‘𝐾) |
12 | 7, 11 | tsettps 22763 | . 2 ⊢ ({∅, 𝐴} ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp) |
13 | 1, 2, 12 | mp2b 10 | 1 ⊢ 𝐾 ∈ TopSp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ∈ wcel 2105 Vcvv 3473 ∅c0 4322 {cpr 4630 〈cop 4634 ‘cfv 6543 ndxcnx 17133 Basecbs 17151 TopSetcts 17210 TopOnctopon 22732 TopSpctps 22754 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-uz 12830 df-fz 13492 df-struct 17087 df-slot 17122 df-ndx 17134 df-base 17152 df-tset 17223 df-rest 17375 df-topn 17376 df-top 22716 df-topon 22733 df-topsp 22755 |
This theorem is referenced by: (None) |
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