![]() |
Mathbox for Rohan Ridenour |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > mnuss2d | Structured version Visualization version GIF version |
Description: mnussd 42344 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 482 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝐴 ⊆ 𝑥)) → 𝑈 ∈ 𝑀) |
5 | simprl 770 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝐴 ⊆ 𝑥)) → 𝑥 ∈ 𝑈) | |
6 | simprr 772 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝐴 ⊆ 𝑥)) → 𝐴 ⊆ 𝑥) | |
7 | 2, 4, 5, 6 | mnussd 42344 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝐴 ⊆ 𝑥)) → 𝐴 ∈ 𝑈) |
8 | 1, 7 | rexlimddv 3157 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∀wal 1540 = wceq 1542 ∈ wcel 2107 {cab 2715 ∀wral 3063 ∃wrex 3072 ⊆ wss 3909 𝒫 cpw 4559 ∪ cuni 4864 |
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 2709 ax-sep 5255 |
This theorem depends on definitions: df-bi 206 df-an 398 df-3an 1090 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-in 3916 df-ss 3926 df-pw 4561 df-uni 4865 |
This theorem is referenced by: mnupwd 42348 mnuunid 42358 mnurndlem2 42363 |
Copyright terms: Public domain | W3C validator |