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Mirrors > Home > MPE Home > Th. List > Mathboxes > mnuss2d | Structured version Visualization version GIF version |
Description: mnussd 40673 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 483 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝐴 ⊆ 𝑥)) → 𝑈 ∈ 𝑀) |
5 | simprl 769 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝐴 ⊆ 𝑥)) → 𝑥 ∈ 𝑈) | |
6 | simprr 771 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝐴 ⊆ 𝑥)) → 𝐴 ⊆ 𝑥) | |
7 | 2, 4, 5, 6 | mnussd 40673 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝐴 ⊆ 𝑥)) → 𝐴 ∈ 𝑈) |
8 | 1, 7 | rexlimddv 3290 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1534 = wceq 1536 ∈ wcel 2113 {cab 2798 ∀wral 3137 ∃wrex 3138 ⊆ wss 3933 𝒫 cpw 4536 ∪ cuni 4835 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5200 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3495 df-in 3940 df-ss 3949 df-pw 4538 df-uni 4836 |
This theorem is referenced by: mnupwd 40677 mnuunid 40687 mnurndlem2 40692 |
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