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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 | ⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉} |
1strwun.u | ⊢ (𝜑 → 𝑈 ∈ WUni) |
1strwunbndx.b | ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈) |
Ref | Expression |
---|---|
1strwunbndx | ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 𝐺 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1str.g | . 2 ⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉} | |
2 | 1strwun.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
3 | 2 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 𝑈 ∈ WUni) |
4 | 1strwunbndx.b | . . . . 5 ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈) | |
5 | 4 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → (Base‘ndx) ∈ 𝑈) |
6 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 𝐵 ∈ 𝑈) | |
7 | 3, 5, 6 | wunop 10760 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 〈(Base‘ndx), 𝐵〉 ∈ 𝑈) |
8 | 3, 7 | wunsn 10754 | . 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → {〈(Base‘ndx), 𝐵〉} ∈ 𝑈) |
9 | 1, 8 | eqeltrid 2843 | 1 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 𝐺 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {csn 4631 〈cop 4637 ‘cfv 6563 WUnicwun 10738 ndxcnx 17227 Basecbs 17245 |
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-ext 2706 |
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-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-tr 5266 df-wun 10740 |
This theorem is referenced by: 1strwun 17265 1strwunOLD 17266 equivestrcsetc 18208 |
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