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Theorem 1strwunbndx 17266
Description: A constructed one-slot structure in a weak universe containing the index of the base set extractor. (Contributed by AV, 27-Mar-2020.)
Hypotheses
Ref Expression
1str.g 𝐺 = {⟨(Base‘ndx), 𝐵⟩}
1strwun.u (𝜑𝑈 ∈ WUni)
1strwunbndx.b (𝜑 → (Base‘ndx) ∈ 𝑈)
Assertion
Ref Expression
1strwunbndx ((𝜑𝐵𝑈) → 𝐺𝑈)

Proof of Theorem 1strwunbndx
StepHypRef Expression
1 1str.g . 2 𝐺 = {⟨(Base‘ndx), 𝐵⟩}
2 1strwun.u . . . 4 (𝜑𝑈 ∈ WUni)
32adantr 480 . . 3 ((𝜑𝐵𝑈) → 𝑈 ∈ WUni)
4 1strwunbndx.b . . . . 5 (𝜑 → (Base‘ndx) ∈ 𝑈)
54adantr 480 . . . 4 ((𝜑𝐵𝑈) → (Base‘ndx) ∈ 𝑈)
6 simpr 484 . . . 4 ((𝜑𝐵𝑈) → 𝐵𝑈)
73, 5, 6wunop 10763 . . 3 ((𝜑𝐵𝑈) → ⟨(Base‘ndx), 𝐵⟩ ∈ 𝑈)
83, 7wunsn 10757 . 2 ((𝜑𝐵𝑈) → {⟨(Base‘ndx), 𝐵⟩} ∈ 𝑈)
91, 8eqeltrid 2844 1 ((𝜑𝐵𝑈) → 𝐺𝑈)
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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