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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 476 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 𝐵 ∈ 𝑈) | |
7 | 3, 5, 6 | wunop 9423 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 〈(Base‘ndx), 𝐵〉 ∈ 𝑈) |
8 | 3, 7 | wunsn 9417 | . 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → {〈(Base‘ndx), 𝐵〉} ∈ 𝑈) |
9 | 1, 8 | syl5eqel 2692 | 1 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 𝐺 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {csn 4125 〈cop 4131 ‘cfv 5804 WUnicwun 9401 ndxcnx 15692 Basecbs 15695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-tr 4681 df-wun 9403 |
This theorem is referenced by: 1strwun 15808 equivestrcsetc 16615 |
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