Theorem mnuprd 45450

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 486	. . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝑈 ∈ 𝑀)
4		mnuprd.4	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑈)
5	4	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐵 ∈ 𝑈)
6		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐴 = ∅)
7		0ss 4353	. . . 4 ⊢ ∅ ⊆ 𝐵
8	6, 7	eqsstrdi 3957	. . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐴 ⊆ 𝐵)
9		ssidd 3936	. . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐵 ⊆ 𝐵)
10	1, 3, 5, 8, 9	mnuprssd 45443	. 2 ⊢ ((𝜑 ∧ 𝐴 = ∅) → {𝐴, 𝐵} ∈ 𝑈)
11		eqid 2763	. . 3 ⊢ {{∅, {𝐴}}, {{∅}, {𝐵}}} = {{∅, {𝐴}}, {{∅}, {𝐵}}}
12	2	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝑈 ∈ 𝑀)
13		mnuprd.3	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
14	13	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝐴 ∈ 𝑈)
15	4	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝐵 ∈ 𝑈)
16		simpr 490	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → ¬ 𝐴 = ∅)
17	1, 11, 12, 14, 15, 16	mnuprdlem4 45449	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → {𝐴, 𝐵} ∈ 𝑈)
18	10, 17	pm2.61dan 825	1 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900  ∅c0 4281  𝒫 cpw 4560  {csn 4588  {cpr 4590  ∪ cuni 4870

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

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-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-pw 4562  df-sn 4589  df-pr 4591  df-uni 4871

This theorem is used by:  mnuund  45452  mnurndlem2  45456  mnugrud  45458