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Theorem axprlem1 5292
 Description: Lemma for axpr 5297. There exists a set to which all empty sets belong. (Contributed by Rohan Ridenour, 10-Aug-2023.) (Revised by BJ, 13-Aug-2023.)
Assertion
Ref Expression
axprlem1 𝑥𝑦(∀𝑧 ¬ 𝑧𝑦𝑦𝑥)
Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem axprlem1
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ax-pow 5234 . . 3 𝑥𝑦(∀𝑧(𝑧𝑦𝑧𝑤) → 𝑦𝑥)
2 pm2.21 123 . . . . . . . 8 𝑧𝑦 → (𝑧𝑦𝑧𝑤))
32alimi 1813 . . . . . . 7 (∀𝑧 ¬ 𝑧𝑦 → ∀𝑧(𝑧𝑦𝑧𝑤))
43a1i 11 . . . . . 6 (∀𝑧 ¬ 𝑧𝑤 → (∀𝑧 ¬ 𝑧𝑦 → ∀𝑧(𝑧𝑦𝑧𝑤)))
54imim1d 82 . . . . 5 (∀𝑧 ¬ 𝑧𝑤 → ((∀𝑧(𝑧𝑦𝑧𝑤) → 𝑦𝑥) → (∀𝑧 ¬ 𝑧𝑦𝑦𝑥)))
65alimdv 1917 . . . 4 (∀𝑧 ¬ 𝑧𝑤 → (∀𝑦(∀𝑧(𝑧𝑦𝑧𝑤) → 𝑦𝑥) → ∀𝑦(∀𝑧 ¬ 𝑧𝑦𝑦𝑥)))
76eximdv 1918 . . 3 (∀𝑧 ¬ 𝑧𝑤 → (∃𝑥𝑦(∀𝑧(𝑧𝑦𝑧𝑤) → 𝑦𝑥) → ∃𝑥𝑦(∀𝑧 ¬ 𝑧𝑦𝑦𝑥)))
81, 7mpi 20 . 2 (∀𝑧 ¬ 𝑧𝑤 → ∃𝑥𝑦(∀𝑧 ¬ 𝑧𝑦𝑦𝑥))
9 ax-nul 5177 . 2 𝑤𝑧 ¬ 𝑧𝑤
108, 9exlimiiv 1932 1 𝑥𝑦(∀𝑧 ¬ 𝑧𝑦𝑦𝑥)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1536  ∃wex 1781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-nul 5177  ax-pow 5234 This theorem depends on definitions:  df-bi 210  df-ex 1782 This theorem is referenced by:  axprlem2  5293  axprlem4  5295
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