Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mnuprssd | Structured version Visualization version GIF version |
Description: A minimal universe contains pairs of subsets of an element of the universe. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
Ref | Expression |
---|---|
mnuprssd.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))))} |
mnuprssd.2 | ⊢ (𝜑 → 𝑈 ∈ 𝑀) |
mnuprssd.3 | ⊢ (𝜑 → 𝐶 ∈ 𝑈) |
mnuprssd.4 | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
mnuprssd.5 | ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Ref | Expression |
---|---|
mnuprssd | ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnuprssd.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 | mnuprssd.2 | . 2 ⊢ (𝜑 → 𝑈 ∈ 𝑀) | |
3 | mnuprssd.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑈) | |
4 | 1, 2, 3 | mnupwd 40652 | . 2 ⊢ (𝜑 → 𝒫 𝐶 ∈ 𝑈) |
5 | mnuprssd.4 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | |
6 | 3, 5 | sselpwd 5230 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐶) |
7 | mnuprssd.5 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | |
8 | 3, 7 | sselpwd 5230 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐶) |
9 | 6, 8 | prssd 4755 | . 2 ⊢ (𝜑 → {𝐴, 𝐵} ⊆ 𝒫 𝐶) |
10 | 1, 2, 4, 9 | mnussd 40648 | 1 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1535 = wceq 1537 ∈ wcel 2114 {cab 2799 ∀wral 3138 ∃wrex 3139 ⊆ wss 3936 𝒫 cpw 4539 {cpr 4569 ∪ cuni 4838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-pw 4541 df-sn 4568 df-pr 4570 df-uni 4839 |
This theorem is referenced by: mnuprss2d 40655 mnuprd 40661 |
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