Step |
Hyp |
Ref |
Expression |
1 |
|
elex |
⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ V ) |
2 |
|
prssi |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → { 𝑥 , 𝑦 } ⊆ 𝐴 ) |
3 |
|
df2o3 |
⊢ 2o = { ∅ , 1o } |
4 |
|
0ex |
⊢ ∅ ∈ V |
5 |
4
|
a1i |
⊢ ( 𝑥 ≠ 𝑦 → ∅ ∈ V ) |
6 |
|
1oex |
⊢ 1o ∈ V |
7 |
6
|
a1i |
⊢ ( 𝑥 ≠ 𝑦 → 1o ∈ V ) |
8 |
|
vex |
⊢ 𝑥 ∈ V |
9 |
8
|
a1i |
⊢ ( 𝑥 ≠ 𝑦 → 𝑥 ∈ V ) |
10 |
|
vex |
⊢ 𝑦 ∈ V |
11 |
10
|
a1i |
⊢ ( 𝑥 ≠ 𝑦 → 𝑦 ∈ V ) |
12 |
|
1n0 |
⊢ 1o ≠ ∅ |
13 |
12
|
necomi |
⊢ ∅ ≠ 1o |
14 |
13
|
a1i |
⊢ ( 𝑥 ≠ 𝑦 → ∅ ≠ 1o ) |
15 |
|
id |
⊢ ( 𝑥 ≠ 𝑦 → 𝑥 ≠ 𝑦 ) |
16 |
5 7 9 11 14 15
|
en2prd |
⊢ ( 𝑥 ≠ 𝑦 → { ∅ , 1o } ≈ { 𝑥 , 𝑦 } ) |
17 |
3 16
|
eqbrtrid |
⊢ ( 𝑥 ≠ 𝑦 → 2o ≈ { 𝑥 , 𝑦 } ) |
18 |
|
endom |
⊢ ( 2o ≈ { 𝑥 , 𝑦 } → 2o ≼ { 𝑥 , 𝑦 } ) |
19 |
17 18
|
syl |
⊢ ( 𝑥 ≠ 𝑦 → 2o ≼ { 𝑥 , 𝑦 } ) |
20 |
|
domssr |
⊢ ( ( 𝐴 ∈ V ∧ { 𝑥 , 𝑦 } ⊆ 𝐴 ∧ 2o ≼ { 𝑥 , 𝑦 } ) → 2o ≼ 𝐴 ) |
21 |
20
|
3expib |
⊢ ( 𝐴 ∈ V → ( ( { 𝑥 , 𝑦 } ⊆ 𝐴 ∧ 2o ≼ { 𝑥 , 𝑦 } ) → 2o ≼ 𝐴 ) ) |
22 |
2 19 21
|
syl2ani |
⊢ ( 𝐴 ∈ V → ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑥 ≠ 𝑦 ) → 2o ≼ 𝐴 ) ) |
23 |
22
|
expd |
⊢ ( 𝐴 ∈ V → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → ( 𝑥 ≠ 𝑦 → 2o ≼ 𝐴 ) ) ) |
24 |
23
|
rexlimdvv |
⊢ ( 𝐴 ∈ V → ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐴 𝑥 ≠ 𝑦 → 2o ≼ 𝐴 ) ) |
25 |
1 24
|
syl |
⊢ ( 𝐴 ∈ 𝑉 → ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐴 𝑥 ≠ 𝑦 → 2o ≼ 𝐴 ) ) |
26 |
25
|
imp |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐴 𝑥 ≠ 𝑦 ) → 2o ≼ 𝐴 ) |