Theorem 1strwunbndx 16592

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 484	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 𝑈 ∈ WUni)
4		1strwunbndx.b	. . . . 5 ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈)
5	4	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → (Base‘ndx) ∈ 𝑈)
6		simpr 488	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 𝐵 ∈ 𝑈)
7	3, 5, 6	wunop 10133	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → ⟨(Base‘ndx), 𝐵⟩ ∈ 𝑈)
8	3, 7	wunsn 10127	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → {⟨(Base‘ndx), 𝐵⟩} ∈ 𝑈)
9	1, 8	eqeltrid 2894	1 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 𝐺 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {csn 4525  ⟨cop 4531  ‘cfv 6324  WUnicwun 10111  ndxcnx 16472  Basecbs 16475

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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

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 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-tr 5137  df-wun 10113

This theorem is referenced by:  1strwun  16593  equivestrcsetc  17394