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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 483 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 𝑈 ∈ WUni) |
4 | 1strwunbndx.b | . . . . 5 ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈) | |
5 | 4 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → (Base‘ndx) ∈ 𝑈) |
6 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 𝐵 ∈ 𝑈) | |
7 | 3, 5, 6 | wunop 10146 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 〈(Base‘ndx), 𝐵〉 ∈ 𝑈) |
8 | 3, 7 | wunsn 10140 | . 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → {〈(Base‘ndx), 𝐵〉} ∈ 𝑈) |
9 | 1, 8 | eqeltrid 2919 | 1 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 𝐺 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {csn 4569 〈cop 4575 ‘cfv 6357 WUnicwun 10124 ndxcnx 16482 Basecbs 16485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-tr 5175 df-wun 10126 |
This theorem is referenced by: 1strwun 16603 equivestrcsetc 17404 |
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