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	⊢ (𝜑 → 𝑈 ∈ WUni)
1strwunbndx.b	⊢ (𝜑 → (Base‘ndx) ∈ 𝑈)

Assertion

Ref	Expression
1strwunbndx	⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 𝐺 ∈ 𝑈)

Proof of Theorem 1strwunbndx

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 484	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 𝐵 ∈ 𝑈)
7	3, 5, 6	wunop 10763	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → ⟨(Base‘ndx), 𝐵⟩ ∈ 𝑈)
8	3, 7	wunsn 10757	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → {⟨(Base‘ndx), 𝐵⟩} ∈ 𝑈)
9	1, 8	eqeltrid 2844	1 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 𝐺 ∈ 𝑈)

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