Theorem 1strwunbndx 17154

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 10635	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → ⟨(Base‘ndx), 𝐵⟩ ∈ 𝑈)
8	3, 7	wunsn 10629	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → {⟨(Base‘ndx), 𝐵⟩} ∈ 𝑈)
9	1, 8	eqeltrid 2840	1 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 𝐺 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {csn 4580  ⟨cop 4586  ‘cfv 6492  WUnicwun 10613  ndxcnx 17122  Basecbs 17138

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-tr 5206  df-wun 10615

This theorem is referenced by:  1strwun  17155  equivestrcsetc  18077