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Mirrors > Home > MPE Home > Th. List > 1strwunbndx | Structured version Visualization version GIF version |
Description: A constructed one-slot structure in a weak universe containing the index of the base set extractor. (Contributed by AV, 27-Mar-2020.) |
Ref | Expression |
---|---|
1str.g | β’ πΊ = {β¨(Baseβndx), π΅β©} |
1strwun.u | β’ (π β π β WUni) |
1strwunbndx.b | β’ (π β (Baseβndx) β π) |
Ref | Expression |
---|---|
1strwunbndx | β’ ((π β§ π΅ β π) β πΊ β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1str.g | . 2 β’ πΊ = {β¨(Baseβndx), π΅β©} | |
2 | 1strwun.u | . . . 4 β’ (π β π β WUni) | |
3 | 2 | adantr 482 | . . 3 β’ ((π β§ π΅ β π) β π β WUni) |
4 | 1strwunbndx.b | . . . . 5 β’ (π β (Baseβndx) β π) | |
5 | 4 | adantr 482 | . . . 4 β’ ((π β§ π΅ β π) β (Baseβndx) β π) |
6 | simpr 486 | . . . 4 β’ ((π β§ π΅ β π) β π΅ β π) | |
7 | 3, 5, 6 | wunop 10663 | . . 3 β’ ((π β§ π΅ β π) β β¨(Baseβndx), π΅β© β π) |
8 | 3, 7 | wunsn 10657 | . 2 β’ ((π β§ π΅ β π) β {β¨(Baseβndx), π΅β©} β π) |
9 | 1, 8 | eqeltrid 2838 | 1 β’ ((π β§ π΅ β π) β πΊ β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 {csn 4587 β¨cop 4593 βcfv 6497 WUnicwun 10641 ndxcnx 17070 Basecbs 17088 |
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-ext 2704 |
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-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-tr 5224 df-wun 10643 |
This theorem is referenced by: 1strwun 17108 1strwunOLD 17109 equivestrcsetc 18045 |
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