Theorem mnuprssd 44301

Description: A minimal universe contains pairs of subsets of an element of the universe. (Contributed by Rohan Ridenour, 13-Aug-2023.)

Hypotheses

Ref	Expression
mnuprssd.1	⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))}
mnuprssd.2	⊢ (𝜑 → 𝑈 ∈ 𝑀)
mnuprssd.3	⊢ (𝜑 → 𝐶 ∈ 𝑈)
mnuprssd.4	⊢ (𝜑 → 𝐴 ⊆ 𝐶)
mnuprssd.5	⊢ (𝜑 → 𝐵 ⊆ 𝐶)

Assertion

Ref	Expression
mnuprssd	⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈)

Distinct variable groups:   𝑈,𝑘,𝑚,𝑛,𝑟,𝑝,𝑙   𝑈,𝑞,𝑘,𝑚,𝑛,𝑝,𝑙

Allowed substitution hints:   𝜑(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝐴(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝐵(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝐶(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝑀(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem mnuprssd

Step	Hyp	Ref	Expression
1		mnuprssd.1	. 2 ⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))}
2		mnuprssd.2	. 2 ⊢ (𝜑 → 𝑈 ∈ 𝑀)
3		mnuprssd.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
4	1, 2, 3	mnupwd 44299	. 2 ⊢ (𝜑 → 𝒫 𝐶 ∈ 𝑈)
5		mnuprssd.4	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
6	3, 5	sselpwd 5266	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐶)
7		mnuprssd.5	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
8	3, 7	sselpwd 5266	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐶)
9	6, 8	prssd 4774	. 2 ⊢ (𝜑 → {𝐴, 𝐵} ⊆ 𝒫 𝐶)
10	1, 2, 4, 9	mnussd 44295	1 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1539   = wceq 1541   ∈ wcel 2111  {cab 2709  ∀wral 3047  ∃wrex 3056   ⊆ wss 3902  𝒫 cpw 4550  {cpr 4578  ∪ cuni 4859

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 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244

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 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-pw 4552  df-sn 4577  df-pr 4579  df-uni 4860

This theorem is referenced by:  mnuprss2d  44302  mnuprd  44308