Theorem mnuss2d 45015

Description: mnussd 45014 with arguments provided with an existential quantifier. (Contributed by Rohan Ridenour, 13-Aug-2023.)

Hypotheses

Ref	Expression
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

Ref	Expression
mnuss2d	⊢ (𝜑 → 𝐴 ∈ 𝑈)

Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝑈   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)   𝑀(𝑥, 𝑘, 𝑚, 𝑛, 𝑟, 𝑞, 𝑝, 𝑙)

Proof of Theorem mnuss2d

Step	Hyp	Ref	Expression
1		mnuss2d.3	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑈 𝐴 ⊆ 𝑥)
2		mnuss2d.1	. . 3 ⊢ 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))))}
3		mnuss2d.2	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑀)
4	3	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝐴 ⊆ 𝑥)) → 𝑈 ∈ 𝑀)
5		simprl 783	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝐴 ⊆ 𝑥)) → 𝑥 ∈ 𝑈)
6		simprr 785	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝐴 ⊆ 𝑥)) → 𝐴 ⊆ 𝑥)
7	2, 4, 5, 6	mnussd 45014	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝐴 ⊆ 𝑥)) → 𝐴 ∈ 𝑈)
8	1, 7	rexlimddv 3175	1 ⊢ (𝜑 → 𝐴 ∈ 𝑈)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2146  {cab 2744  ∀wral 3082  ∃wrex 3092   ⊆ wss 3908  𝒫 cpw 4567  ∪ cuni 4877

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-in 3915  df-ss 3925  df-pw 4569  df-uni 4878

This theorem is used by:  mnupwd  45018  mnuunid  45028  mnurndlem2  45033