Theorem mnuprss2d 45021

Description: Special case of mnuprssd 45020. (Contributed by Rohan Ridenour, 13-Aug-2023.)

Hypotheses

Ref	Expression
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

Ref	Expression
mnuprss2d	⊢ (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 mnuprss2d

Step	Hyp	Ref	Expression
1		mnuprss2d.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		mnuprss2d.2	. 2 ⊢ (𝜑 → 𝑈 ∈ 𝑀)
3		mnuprss2d.3	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
4		mnuprss2d.4	. . 3 ⊢ 𝐴 ⊆ 𝐶
5	4	a1i 11	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
6		mnuprss2d.5	. . 3 ⊢ 𝐵 ⊆ 𝐶
7	6	a1i 11	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
8	1, 2, 3, 5, 7	mnuprssd 45020	1 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 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  {cpr 4596  ∪ 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  ax-nul 5274

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  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-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-pw 4569  df-sn 4595  df-pr 4597  df-uni 4878

This theorem is used by:  mnuprdlem4  45026