Theorem mnuprd 44857

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 484	. . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝑈 ∈ 𝑀)
4		mnuprd.4	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑈)
5	4	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐵 ∈ 𝑈)
6		simpr 488	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐴 = ∅)
7		0ss 4356	. . . 4 ⊢ ∅ ⊆ 𝐵
8	6, 7	eqsstrdi 3982	. . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐴 ⊆ 𝐵)
9		ssidd 3961	. . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐵 ⊆ 𝐵)
10	1, 3, 5, 8, 9	mnuprssd 44850	. 2 ⊢ ((𝜑 ∧ 𝐴 = ∅) → {𝐴, 𝐵} ∈ 𝑈)
11		eqid 2764	. . 3 ⊢ {{∅, {𝐴}}, {{∅}, {𝐵}}} = {{∅, {𝐴}}, {{∅}, {𝐵}}}
12	2	adantr 484	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝑈 ∈ 𝑀)
13		mnuprd.3	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
14	13	adantr 484	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝐴 ∈ 𝑈)
15	4	adantr 484	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝐵 ∈ 𝑈)
16		simpr 488	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → ¬ 𝐴 = ∅)
17	1, 11, 12, 14, 15, 16	mnuprdlem4 44856	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → {𝐴, 𝐵} ∈ 𝑈)
18	10, 17	pm2.61dan 822	1 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1560   = wceq 1562   ∈ wcel 2144  {cab 2742  ∀wral 3078  ∃wrex 3088   ⊆ wss 3906  ∅c0 4287  𝒫 cpw 4557  {csn 4584  {cpr 4586  ∪ cuni 4867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-pw 4559  df-sn 4585  df-pr 4587  df-uni 4868

This theorem is referenced by:  mnuund  44859  mnurndlem2  44863  mnugrud  44865