Step |
Hyp |
Ref |
Expression |
1 |
|
exel |
⊢ ∃ 𝑥 ∃ 𝑧 𝑧 ∈ 𝑥 |
2 |
|
ax-nul |
⊢ ∃ 𝑦 ∀ 𝑧 ¬ 𝑧 ∈ 𝑦 |
3 |
|
exdistrv |
⊢ ( ∃ 𝑥 ∃ 𝑦 ( ∃ 𝑧 𝑧 ∈ 𝑥 ∧ ∀ 𝑧 ¬ 𝑧 ∈ 𝑦 ) ↔ ( ∃ 𝑥 ∃ 𝑧 𝑧 ∈ 𝑥 ∧ ∃ 𝑦 ∀ 𝑧 ¬ 𝑧 ∈ 𝑦 ) ) |
4 |
1 2 3
|
mpbir2an |
⊢ ∃ 𝑥 ∃ 𝑦 ( ∃ 𝑧 𝑧 ∈ 𝑥 ∧ ∀ 𝑧 ¬ 𝑧 ∈ 𝑦 ) |
5 |
|
ax9v1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦 ) ) |
6 |
5
|
eximdv |
⊢ ( 𝑥 = 𝑦 → ( ∃ 𝑧 𝑧 ∈ 𝑥 → ∃ 𝑧 𝑧 ∈ 𝑦 ) ) |
7 |
|
df-ex |
⊢ ( ∃ 𝑧 𝑧 ∈ 𝑦 ↔ ¬ ∀ 𝑧 ¬ 𝑧 ∈ 𝑦 ) |
8 |
6 7
|
imbitrdi |
⊢ ( 𝑥 = 𝑦 → ( ∃ 𝑧 𝑧 ∈ 𝑥 → ¬ ∀ 𝑧 ¬ 𝑧 ∈ 𝑦 ) ) |
9 |
8
|
com12 |
⊢ ( ∃ 𝑧 𝑧 ∈ 𝑥 → ( 𝑥 = 𝑦 → ¬ ∀ 𝑧 ¬ 𝑧 ∈ 𝑦 ) ) |
10 |
9
|
con2d |
⊢ ( ∃ 𝑧 𝑧 ∈ 𝑥 → ( ∀ 𝑧 ¬ 𝑧 ∈ 𝑦 → ¬ 𝑥 = 𝑦 ) ) |
11 |
10
|
imp |
⊢ ( ( ∃ 𝑧 𝑧 ∈ 𝑥 ∧ ∀ 𝑧 ¬ 𝑧 ∈ 𝑦 ) → ¬ 𝑥 = 𝑦 ) |
12 |
11
|
2eximi |
⊢ ( ∃ 𝑥 ∃ 𝑦 ( ∃ 𝑧 𝑧 ∈ 𝑥 ∧ ∀ 𝑧 ¬ 𝑧 ∈ 𝑦 ) → ∃ 𝑥 ∃ 𝑦 ¬ 𝑥 = 𝑦 ) |
13 |
4 12
|
ax-mp |
⊢ ∃ 𝑥 ∃ 𝑦 ¬ 𝑥 = 𝑦 |