Theorem mnuprss2d

Description: Special case of mnuprssd . (Contributed by Rohan Ridenour, 13-Aug-2023)

	Ref	Expression
Hypotheses	mnuprss2d.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 ) ) ) ) }
	mnuprss2d.2	⊢ ( 𝜑 → 𝑈 ∈ 𝑀 )
	mnuprss2d.3	⊢ ( 𝜑 → 𝐶 ∈ 𝑈 )
	mnuprss2d.4	⊢ 𝐴 ⊆ 𝐶
	mnuprss2d.5	⊢ 𝐵 ⊆ 𝐶
Assertion	mnuprss2d	⊢ ( 𝜑 → { 𝐴 , 𝐵 } ∈ 𝑈 )

Proof

Step	Hyp	Ref	Expression
1		mnuprss2d.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 ) ) ) ) }
2		mnuprss2d.2	⊢ ( 𝜑 → 𝑈 ∈ 𝑀 )
3		mnuprss2d.3	⊢ ( 𝜑 → 𝐶 ∈ 𝑈 )
4		mnuprss2d.4	⊢ 𝐴 ⊆ 𝐶
5		mnuprss2d.5	⊢ 𝐵 ⊆ 𝐶
6	4	a1i	⊢ ( 𝜑 → 𝐴 ⊆ 𝐶 )
7	5	a1i	⊢ ( 𝜑 → 𝐵 ⊆ 𝐶 )
8	1 2 3 6 7	mnuprssd	⊢ ( 𝜑 → { 𝐴 , 𝐵 } ∈ 𝑈 )