Theorem mnuss2d

Description: mnussd with arguments provided with an existential quantifier. (Contributed by Rohan Ridenour, 13-Aug-2023)

	Ref	Expression
Hypotheses	mnuss2d.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 ) ) ) ) }
	mnuss2d.2	⊢ ( 𝜑 → 𝑈 ∈ 𝑀 )
	mnuss2d.3	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝑈 𝐴 ⊆ 𝑥 )
Assertion	mnuss2d	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )

Proof

Step	Hyp	Ref	Expression
1		mnuss2d.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		mnuss2d.2	⊢ ( 𝜑 → 𝑈 ∈ 𝑀 )
3		mnuss2d.3	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝑈 𝐴 ⊆ 𝑥 )
4	2	adantr	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝐴 ⊆ 𝑥 ) ) → 𝑈 ∈ 𝑀 )
5		simprl	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝐴 ⊆ 𝑥 ) ) → 𝑥 ∈ 𝑈 )
6		simprr	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝐴 ⊆ 𝑥 ) ) → 𝐴 ⊆ 𝑥 )
7	1 4 5 6	mnussd	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑈 ∧ 𝐴 ⊆ 𝑥 ) ) → 𝐴 ∈ 𝑈 )
8	3 7	rexlimddv	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )