Theorem mnuprss2d 44839

Description: Special case of mnuprssd 44838. (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 44838	1 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1561   = wceq 1563   ∈ wcel 2145  {cab 2743  ∀wral 3079  ∃wrex 3089   ⊆ wss 3907  𝒫 cpw 4558  {cpr 4587  ∪ cuni 4867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5250  ax-nul 5260

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-pw 4560  df-sn 4586  df-pr 4588  df-uni 4868

This theorem is referenced by:  mnuprdlem4  44844