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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 6670 | . 2 ⊢ 𝐴 ∈ V |
3 | opnssborel.b | . . 3 ⊢ 𝐵 = (SalGen‘𝐴) | |
4 | 3 | sssalgen 42708 | . 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 2114 Vcvv 3486 ⊆ wss 3924 ‘cfv 6341 TopOpenctopn 16678 ℝ^crrx 23969 SalGencsalgen 42687 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-int 4863 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-iota 6300 df-fun 6343 df-fv 6349 df-salg 42684 df-salgen 42688 |
This theorem is referenced by: (None) |
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