Theorem 1strwunbndx 17170

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 10723	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → ⟨(Base‘ndx), 𝐵⟩ ∈ 𝑈)
8	3, 7	wunsn 10717	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → {⟨(Base‘ndx), 𝐵⟩} ∈ 𝑈)
9	1, 8	eqeltrid 2836	1 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 𝐺 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  {csn 4628  ⟨cop 4634  ‘cfv 6543  WUnicwun 10701  ndxcnx 17133  Basecbs 17151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-tr 5266  df-wun 10703

This theorem is referenced by:  1strwun  17171  1strwunOLD  17172  equivestrcsetc  18114