Theorem mnuprd 41894

Description: Minimal universes are closed under pairing. (Contributed by Rohan Ridenour, 13-Aug-2023.)

Hypotheses

Ref	Expression
mnuprd.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))))}
mnuprd.2	⊢ (𝜑 → 𝑈 ∈ 𝑀)
mnuprd.3	⊢ (𝜑 → 𝐴 ∈ 𝑈)
mnuprd.4	⊢ (𝜑 → 𝐵 ∈ 𝑈)

Assertion

Ref	Expression
mnuprd	⊢ (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)   𝑀(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem mnuprd

Step	Hyp	Ref	Expression
1		mnuprd.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))))}
2		mnuprd.2	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑀)
3	2	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝑈 ∈ 𝑀)
4		mnuprd.4	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑈)
5	4	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐵 ∈ 𝑈)
6		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐴 = ∅)
7		0ss 4330	. . . 4 ⊢ ∅ ⊆ 𝐵
8	6, 7	eqsstrdi 3975	. . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐴 ⊆ 𝐵)
9		ssidd 3944	. . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐵 ⊆ 𝐵)
10	1, 3, 5, 8, 9	mnuprssd 41887	. 2 ⊢ ((𝜑 ∧ 𝐴 = ∅) → {𝐴, 𝐵} ∈ 𝑈)
11		eqid 2738	. . 3 ⊢ {{∅, {𝐴}}, {{∅}, {𝐵}}} = {{∅, {𝐴}}, {{∅}, {𝐵}}}
12	2	adantr 481	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝑈 ∈ 𝑀)
13		mnuprd.3	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
14	13	adantr 481	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝐴 ∈ 𝑈)
15	4	adantr 481	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝐵 ∈ 𝑈)
16		simpr 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → ¬ 𝐴 = ∅)
17	1, 11, 12, 14, 15, 16	mnuprdlem4 41893	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → {𝐴, 𝐵} ∈ 𝑈)
18	10, 17	pm2.61dan 810	1 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  ∀wal 1537   = wceq 1539   ∈ wcel 2106  {cab 2715  ∀wral 3064  ∃wrex 3065   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  {csn 4561  {cpr 4563  ∪ cuni 4839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-pw 4535  df-sn 4562  df-pr 4564  df-uni 4840

This theorem is referenced by:  mnuund  41896  mnurndlem2  41900  mnugrud  41902