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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 6677 | . 2 ⊢ 𝐴 ∈ V |
3 | opnssborel.b | . . 3 ⊢ 𝐵 = (SalGen‘𝐴) | |
4 | 3 | sssalgen 42495 | . 2 ⊢ (𝐴 ∈ V → 𝐴 ⊆ 𝐵) |
5 | 2, 4 | ax-mp 5 | 1 ⊢ 𝐴 ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 Vcvv 3492 ⊆ wss 3933 ‘cfv 6348 TopOpenctopn 16683 ℝ^crrx 23913 SalGencsalgen 42474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-int 4868 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-salg 42471 df-salgen 42475 |
This theorem is referenced by: (None) |
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