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Theorem 1strwunbndx 17264
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 10760 . . 3 ((𝜑𝐵𝑈) → ⟨(Base‘ndx), 𝐵⟩ ∈ 𝑈)
83, 7wunsn 10754 . 2 ((𝜑𝐵𝑈) → {⟨(Base‘ndx), 𝐵⟩} ∈ 𝑈)
91, 8eqeltrid 2843 1 ((𝜑𝐵𝑈) → 𝐺𝑈)
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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