Theorem 1strwunbndx 17166

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 10647	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → ⟨(Base‘ndx), 𝐵⟩ ∈ 𝑈)
8	3, 7	wunsn 10641	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → {⟨(Base‘ndx), 𝐵⟩} ∈ 𝑈)
9	1, 8	eqeltrid 2841	1 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 𝐺 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {csn 4582  ⟨cop 4588  ‘cfv 6502  WUnicwun 10625  ndxcnx 17134  Basecbs 17150

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-tr 5208  df-wun 10627

This theorem is referenced by:  1strwun  17167  equivestrcsetc  18089