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Mathbox for Glauco Siliprandi |
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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 π is not required to be a set). (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
Ref | Expression |
---|---|
opnssborel.a | β’ π΄ = (TopOpenβ(β^βπ)) |
opnssborel.b | β’ π΅ = (SalGenβπ΄) |
Ref | Expression |
---|---|
opnssborel | β’ π΄ β π΅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opnssborel.a | . . 3 β’ π΄ = (TopOpenβ(β^βπ)) | |
2 | 1 | fvexi 6860 | . 2 β’ π΄ β V |
3 | opnssborel.b | . . 3 β’ π΅ = (SalGenβπ΄) | |
4 | 3 | sssalgen 44666 | . 2 β’ (π΄ β V β π΄ β π΅) |
5 | 2, 4 | ax-mp 5 | 1 β’ π΄ β π΅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 β wcel 2107 Vcvv 3447 β wss 3914 βcfv 6500 TopOpenctopn 17311 β^crrx 24770 SalGencsalgen 44643 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-iota 6452 df-fun 6502 df-fv 6508 df-salg 44640 df-salgen 44644 |
This theorem is referenced by: (None) |
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