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Mirrors > Home > MPE Home > Th. List > Mathboxes > mnutrcld | Structured version Visualization version GIF version |
Description: Minimal universes contain the elements of their elements. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
Ref | Expression |
---|---|
mnutrcld.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))))} |
mnutrcld.2 | ⊢ (𝜑 → 𝑈 ∈ 𝑀) |
mnutrcld.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
mnutrcld.4 | ⊢ (𝜑 → 𝐵 ∈ 𝐴) |
Ref | Expression |
---|---|
mnutrcld | ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnutrcld.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 | mnutrcld.2 | . 2 ⊢ (𝜑 → 𝑈 ∈ 𝑀) | |
3 | mnutrcld.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
4 | 1, 2, 3 | mnuunid 40686 | . 2 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈) |
5 | mnutrcld.4 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝐴) | |
6 | elssuni 4854 | . . 3 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝐴) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝜑 → 𝐵 ⊆ ∪ 𝐴) |
8 | 1, 2, 4, 7 | mnussd 40672 | 1 ⊢ (𝜑 → 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 3924 𝒫 cpw 4525 ∪ cuni 4824 |
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 5189 |
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 3488 df-sbc 3764 df-in 3931 df-ss 3940 df-pw 4527 df-sn 4554 df-uni 4825 |
This theorem is referenced by: mnutrd 40689 mnurndlem2 40691 |
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