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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mnusnd | Structured version Visualization version GIF version |
Description: Minimal universes are closed under singletons. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
Ref | Expression |
---|---|
mnusnd.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))))} |
mnusnd.2 | ⊢ (𝜑 → 𝑈 ∈ 𝑀) |
mnusnd.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
Ref | Expression |
---|---|
mnusnd | ⊢ (𝜑 → {𝐴} ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnusnd.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 | mnusnd.2 | . 2 ⊢ (𝜑 → 𝑈 ∈ 𝑀) | |
3 | mnusnd.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
4 | 1, 2, 3 | mnupwd 42639 | . 2 ⊢ (𝜑 → 𝒫 𝐴 ∈ 𝑈) |
5 | snsspw 4806 | . . 3 ⊢ {𝐴} ⊆ 𝒫 𝐴 | |
6 | 5 | a1i 11 | . 2 ⊢ (𝜑 → {𝐴} ⊆ 𝒫 𝐴) |
7 | 1, 2, 4, 6 | mnussd 42635 | 1 ⊢ (𝜑 → {𝐴} ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∀wal 1540 = wceq 1542 ∈ wcel 2107 {cab 2710 ∀wral 3061 ∃wrex 3070 ⊆ wss 3914 𝒫 cpw 4564 {csn 4590 ∪ cuni 4869 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5260 ax-nul 5267 |
This theorem depends on definitions: df-bi 206 df-an 398 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-in 3921 df-ss 3931 df-nul 4287 df-pw 4566 df-sn 4591 df-uni 4870 |
This theorem is referenced by: mnuprdlem4 42647 |
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