Step |
Hyp |
Ref |
Expression |
1 |
|
dfss2 |
⊢ ( { 𝐴 } ⊆ 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ { 𝐴 } → 𝑥 ∈ 𝐵 ) ) |
2 |
|
idn1 |
⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 ) |
3 |
|
idn2 |
⊢ ( 𝐴 ∈ 𝐵 , 𝑥 ∈ { 𝐴 } ▶ 𝑥 ∈ { 𝐴 } ) |
4 |
|
velsn |
⊢ ( 𝑥 ∈ { 𝐴 } ↔ 𝑥 = 𝐴 ) |
5 |
3 4
|
e2bi |
⊢ ( 𝐴 ∈ 𝐵 , 𝑥 ∈ { 𝐴 } ▶ 𝑥 = 𝐴 ) |
6 |
|
eleq1a |
⊢ ( 𝐴 ∈ 𝐵 → ( 𝑥 = 𝐴 → 𝑥 ∈ 𝐵 ) ) |
7 |
2 5 6
|
e12 |
⊢ ( 𝐴 ∈ 𝐵 , 𝑥 ∈ { 𝐴 } ▶ 𝑥 ∈ 𝐵 ) |
8 |
7
|
in2 |
⊢ ( 𝐴 ∈ 𝐵 ▶ ( 𝑥 ∈ { 𝐴 } → 𝑥 ∈ 𝐵 ) ) |
9 |
8
|
gen11 |
⊢ ( 𝐴 ∈ 𝐵 ▶ ∀ 𝑥 ( 𝑥 ∈ { 𝐴 } → 𝑥 ∈ 𝐵 ) ) |
10 |
|
biimpr |
⊢ ( ( { 𝐴 } ⊆ 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ { 𝐴 } → 𝑥 ∈ 𝐵 ) ) → ( ∀ 𝑥 ( 𝑥 ∈ { 𝐴 } → 𝑥 ∈ 𝐵 ) → { 𝐴 } ⊆ 𝐵 ) ) |
11 |
1 9 10
|
e01 |
⊢ ( 𝐴 ∈ 𝐵 ▶ { 𝐴 } ⊆ 𝐵 ) |
12 |
11
|
in1 |
⊢ ( 𝐴 ∈ 𝐵 → { 𝐴 } ⊆ 𝐵 ) |