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Mirrors > Home > MPE Home > Th. List > Mathboxes > opnssborel | Structured version Visualization version GIF version |
Description: Open sets of a generalized real Euclidean space are Borel sets (notice that this theorem is even more general, because 𝑋 is not required to be a set). (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
Ref | Expression |
---|---|
opnssborel.a | ⊢ 𝐴 = (TopOpen‘(ℝ^‘𝑋)) |
opnssborel.b | ⊢ 𝐵 = (SalGen‘𝐴) |
Ref | Expression |
---|---|
opnssborel | ⊢ 𝐴 ⊆ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opnssborel.a | . . 3 ⊢ 𝐴 = (TopOpen‘(ℝ^‘𝑋)) | |
2 | 1 | fvexi 6921 | . 2 ⊢ 𝐴 ∈ V |
3 | opnssborel.b | . . 3 ⊢ 𝐵 = (SalGen‘𝐴) | |
4 | 3 | sssalgen 46291 | . 2 ⊢ (𝐴 ∈ V → 𝐴 ⊆ 𝐵) |
5 | 2, 4 | ax-mp 5 | 1 ⊢ 𝐴 ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 ‘cfv 6563 TopOpenctopn 17468 ℝ^crrx 25431 SalGencsalgen 46268 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-salg 46265 df-salgen 46269 |
This theorem is referenced by: (None) |
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