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Mirrors > Home > MPE Home > Th. List > axprlem1 | Structured version Visualization version GIF version |
Description: Lemma for axpr 5355. There exists a set to which all empty sets belong. (Contributed by Rohan Ridenour, 10-Aug-2023.) (Revised by BJ, 13-Aug-2023.) |
Ref | Expression |
---|---|
axprlem1 | ⊢ ∃𝑥∀𝑦(∀𝑧 ¬ 𝑧 ∈ 𝑦 → 𝑦 ∈ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-pow 5292 | . . 3 ⊢ ∃𝑥∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤) → 𝑦 ∈ 𝑥) | |
2 | pm2.21 123 | . . . . . . . 8 ⊢ (¬ 𝑧 ∈ 𝑦 → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤)) | |
3 | 2 | alimi 1818 | . . . . . . 7 ⊢ (∀𝑧 ¬ 𝑧 ∈ 𝑦 → ∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤)) |
4 | 3 | a1i 11 | . . . . . 6 ⊢ (∀𝑧 ¬ 𝑧 ∈ 𝑤 → (∀𝑧 ¬ 𝑧 ∈ 𝑦 → ∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤))) |
5 | 4 | imim1d 82 | . . . . 5 ⊢ (∀𝑧 ¬ 𝑧 ∈ 𝑤 → ((∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤) → 𝑦 ∈ 𝑥) → (∀𝑧 ¬ 𝑧 ∈ 𝑦 → 𝑦 ∈ 𝑥))) |
6 | 5 | alimdv 1923 | . . . 4 ⊢ (∀𝑧 ¬ 𝑧 ∈ 𝑤 → (∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤) → 𝑦 ∈ 𝑥) → ∀𝑦(∀𝑧 ¬ 𝑧 ∈ 𝑦 → 𝑦 ∈ 𝑥))) |
7 | 6 | eximdv 1924 | . . 3 ⊢ (∀𝑧 ¬ 𝑧 ∈ 𝑤 → (∃𝑥∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤) → 𝑦 ∈ 𝑥) → ∃𝑥∀𝑦(∀𝑧 ¬ 𝑧 ∈ 𝑦 → 𝑦 ∈ 𝑥))) |
8 | 1, 7 | mpi 20 | . 2 ⊢ (∀𝑧 ¬ 𝑧 ∈ 𝑤 → ∃𝑥∀𝑦(∀𝑧 ¬ 𝑧 ∈ 𝑦 → 𝑦 ∈ 𝑥)) |
9 | ax-nul 5234 | . 2 ⊢ ∃𝑤∀𝑧 ¬ 𝑧 ∈ 𝑤 | |
10 | 8, 9 | exlimiiv 1938 | 1 ⊢ ∃𝑥∀𝑦(∀𝑧 ¬ 𝑧 ∈ 𝑦 → 𝑦 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1540 ∃wex 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-nul 5234 ax-pow 5292 |
This theorem depends on definitions: df-bi 206 df-ex 1787 |
This theorem is referenced by: axprlem2 5351 axprlem4 5353 |
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