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Metamath Proof Explorer
Description: Lemma for the Axiom of Power Sets with no distinct variable conditions.
(Contributed by NM, 4-Jan-2002)
|
|
Ref |
Expression |
|
Assertion |
axpowndlem1 |
⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ¬ 𝑥 = 𝑦 → ∃ 𝑥 ∀ 𝑦 ( ∀ 𝑥 ( ∃ 𝑧 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
pm2.24 |
⊢ ( 𝑥 = 𝑦 → ( ¬ 𝑥 = 𝑦 → ∃ 𝑥 ∀ 𝑦 ( ∀ 𝑥 ( ∃ 𝑧 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ) ) |
| 2 |
1
|
sps |
⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ¬ 𝑥 = 𝑦 → ∃ 𝑥 ∀ 𝑦 ( ∀ 𝑥 ( ∃ 𝑧 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ) ) |