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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mnuss2d | Structured version Visualization version GIF version |
Description: mnussd 44259 with arguments provided with an existential quantifier. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
Ref | Expression |
---|---|
mnuss2d.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))))} |
mnuss2d.2 | ⊢ (𝜑 → 𝑈 ∈ 𝑀) |
mnuss2d.3 | ⊢ (𝜑 → ∃𝑥 ∈ 𝑈 𝐴 ⊆ 𝑥) |
Ref | Expression |
---|---|
mnuss2d | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnuss2d.3 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑈 𝐴 ⊆ 𝑥) | |
2 | mnuss2d.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))))} | |
3 | mnuss2d.2 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑀) | |
4 | 3 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝐴 ⊆ 𝑥)) → 𝑈 ∈ 𝑀) |
5 | simprl 771 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝐴 ⊆ 𝑥)) → 𝑥 ∈ 𝑈) | |
6 | simprr 773 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝐴 ⊆ 𝑥)) → 𝐴 ⊆ 𝑥) | |
7 | 2, 4, 5, 6 | mnussd 44259 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝐴 ⊆ 𝑥)) → 𝐴 ∈ 𝑈) |
8 | 1, 7 | rexlimddv 3159 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1535 = wceq 1537 ∈ wcel 2106 {cab 2712 ∀wral 3059 ∃wrex 3068 ⊆ wss 3963 𝒫 cpw 4605 ∪ 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-ext 2706 ax-sep 5302 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-in 3970 df-ss 3980 df-pw 4607 df-uni 4913 |
This theorem is referenced by: mnupwd 44263 mnuunid 44273 mnurndlem2 44278 |
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