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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 10763 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 〈(Base‘ndx), 𝐵〉 ∈ 𝑈) | 
| 8 | 3, 7 | wunsn 10757 | . 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → {〈(Base‘ndx), 𝐵〉} ∈ 𝑈) | 
| 9 | 1, 8 | eqeltrid 2844 | 1 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 𝐺 ∈ 𝑈) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {csn 4625 〈cop 4631 ‘cfv 6560 WUnicwun 10741 ndxcnx 17231 Basecbs 17248 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-tr 5259 df-wun 10743 | 
| This theorem is referenced by: 1strwun 17267 equivestrcsetc 18198 | 
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