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Mirrors > Home > MPE Home > Th. List > Mathboxes > mnuund | Structured version Visualization version GIF version |
Description: Minimal universes are closed under binary unions. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
Ref | Expression |
---|---|
mnuund.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))))} |
mnuund.2 | ⊢ (𝜑 → 𝑈 ∈ 𝑀) |
mnuund.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
mnuund.4 | ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
Ref | Expression |
---|---|
mnuund | ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnuund.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
2 | mnuund.4 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑈) | |
3 | uniprg 4869 | . . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) | |
4 | 1, 2, 3 | syl2anc 584 | . 2 ⊢ (𝜑 → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) |
5 | mnuund.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))))} | |
6 | mnuund.2 | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑀) | |
7 | 5, 6, 1, 2 | mnuprd 42215 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈) |
8 | 5, 6, 7 | mnuunid 42216 | . 2 ⊢ (𝜑 → ∪ {𝐴, 𝐵} ∈ 𝑈) |
9 | 4, 8 | eqeltrrd 2838 | 1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1538 = wceq 1540 ∈ wcel 2105 {cab 2713 ∀wral 3061 ∃wrex 3070 ∪ cun 3896 ⊆ wss 3898 𝒫 cpw 4547 {cpr 4575 ∪ cuni 4852 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-pw 4549 df-sn 4574 df-pr 4576 df-uni 4853 |
This theorem is referenced by: (None) |
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