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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mnuprd | Structured version Visualization version GIF version |
Description: Minimal universes are closed under pairing. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
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) |
Ref | Expression |
---|---|
mnuprd | ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈) |
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 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝑈 ∈ 𝑀) |
4 | mnuprd.4 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑈) | |
5 | 4 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐵 ∈ 𝑈) |
6 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐴 = ∅) | |
7 | 0ss 4406 | . . . 4 ⊢ ∅ ⊆ 𝐵 | |
8 | 6, 7 | eqsstrdi 4050 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐴 ⊆ 𝐵) |
9 | ssidd 4019 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐵 ⊆ 𝐵) | |
10 | 1, 3, 5, 8, 9 | mnuprssd 44265 | . 2 ⊢ ((𝜑 ∧ 𝐴 = ∅) → {𝐴, 𝐵} ∈ 𝑈) |
11 | eqid 2735 | . . 3 ⊢ {{∅, {𝐴}}, {{∅}, {𝐵}}} = {{∅, {𝐴}}, {{∅}, {𝐵}}} | |
12 | 2 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝑈 ∈ 𝑀) |
13 | mnuprd.3 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
14 | 13 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝐴 ∈ 𝑈) |
15 | 4 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝐵 ∈ 𝑈) |
16 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → ¬ 𝐴 = ∅) | |
17 | 1, 11, 12, 14, 15, 16 | mnuprdlem4 44271 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → {𝐴, 𝐵} ∈ 𝑈) |
18 | 10, 17 | pm2.61dan 813 | 1 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∀wal 1535 = wceq 1537 ∈ wcel 2106 {cab 2712 ∀wral 3059 ∃wrex 3068 ⊆ wss 3963 ∅c0 4339 𝒫 cpw 4605 {csn 4631 {cpr 4633 ∪ cuni 4912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-pw 4607 df-sn 4632 df-pr 4634 df-uni 4913 |
This theorem is referenced by: mnuund 44274 mnurndlem2 44278 mnugrud 44280 |
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