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Theorem 1strwunbndx 16302
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 473 . . 3 ((𝜑𝐵𝑈) → 𝑈 ∈ WUni)
4 1strwunbndx.b . . . . 5 (𝜑 → (Base‘ndx) ∈ 𝑈)
54adantr 473 . . . 4 ((𝜑𝐵𝑈) → (Base‘ndx) ∈ 𝑈)
6 simpr 478 . . . 4 ((𝜑𝐵𝑈) → 𝐵𝑈)
73, 5, 6wunop 9832 . . 3 ((𝜑𝐵𝑈) → ⟨(Base‘ndx), 𝐵⟩ ∈ 𝑈)
83, 7wunsn 9826 . 2 ((𝜑𝐵𝑈) → {⟨(Base‘ndx), 𝐵⟩} ∈ 𝑈)
91, 8syl5eqel 2882 1 ((𝜑𝐵𝑈) → 𝐺𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  {csn 4368  cop 4374  cfv 6101  WUnicwun 9810  ndxcnx 16181  Basecbs 16184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-tr 4946  df-wun 9812
This theorem is referenced by:  1strwun  16303  equivestrcsetc  17107
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