Theorem mnuss2d 40690

Description: mnussd 40689 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 483	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝐴 ⊆ 𝑥)) → 𝑈 ∈ 𝑀)
5		simprl 769	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝐴 ⊆ 𝑥)) → 𝑥 ∈ 𝑈)
6		simprr 771	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝐴 ⊆ 𝑥)) → 𝐴 ⊆ 𝑥)
7	2, 4, 5, 6	mnussd 40689	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝐴 ⊆ 𝑥)) → 𝐴 ∈ 𝑈)
8	1, 7	rexlimddv 3291	1 ⊢ (𝜑 → 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1535   = wceq 1537   ∈ wcel 2114  {cab 2799  ∀wral 3138  ∃wrex 3139   ⊆ wss 3924  𝒫 cpw 4525  ∪ cuni 4824

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5189

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3488  df-in 3931  df-ss 3940  df-pw 4527  df-uni 4825

This theorem is referenced by:  mnupwd  40693  mnuunid  40703  mnurndlem2  40708