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Theorem 1strwunbndx 16591
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 484 . . 3 ((𝜑𝐵𝑈) → 𝑈 ∈ WUni)
4 1strwunbndx.b . . . . 5 (𝜑 → (Base‘ndx) ∈ 𝑈)
54adantr 484 . . . 4 ((𝜑𝐵𝑈) → (Base‘ndx) ∈ 𝑈)
6 simpr 488 . . . 4 ((𝜑𝐵𝑈) → 𝐵𝑈)
73, 5, 6wunop 10133 . . 3 ((𝜑𝐵𝑈) → ⟨(Base‘ndx), 𝐵⟩ ∈ 𝑈)
83, 7wunsn 10127 . 2 ((𝜑𝐵𝑈) → {⟨(Base‘ndx), 𝐵⟩} ∈ 𝑈)
91, 8eqeltrid 2918 1 ((𝜑𝐵𝑈) → 𝐺𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  {csn 4539  cop 4545  cfv 6334  WUnicwun 10111  ndxcnx 16471  Basecbs 16474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-tr 5149  df-wun 10113
This theorem is referenced by:  1strwun  16592  equivestrcsetc  17393
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