Step |
Hyp |
Ref |
Expression |
1 |
|
nosgnn0 |
⊢ ¬ ∅ ∈ { 1o , 2o } |
2 |
1
|
a1i |
⊢ ( ( 𝐴 ∈ No ∧ 𝑋 ∈ On ∧ ( 𝐴 ‘ 𝑋 ) = ∅ ) → ¬ ∅ ∈ { 1o , 2o } ) |
3 |
|
simpl3 |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝑋 ∈ On ∧ ( 𝐴 ‘ 𝑋 ) = ∅ ) ∧ 𝑋 ∈ dom 𝐴 ) → ( 𝐴 ‘ 𝑋 ) = ∅ ) |
4 |
|
simpl1 |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝑋 ∈ On ∧ ( 𝐴 ‘ 𝑋 ) = ∅ ) ∧ 𝑋 ∈ dom 𝐴 ) → 𝐴 ∈ No ) |
5 |
|
norn |
⊢ ( 𝐴 ∈ No → ran 𝐴 ⊆ { 1o , 2o } ) |
6 |
4 5
|
syl |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝑋 ∈ On ∧ ( 𝐴 ‘ 𝑋 ) = ∅ ) ∧ 𝑋 ∈ dom 𝐴 ) → ran 𝐴 ⊆ { 1o , 2o } ) |
7 |
|
nofun |
⊢ ( 𝐴 ∈ No → Fun 𝐴 ) |
8 |
7
|
3ad2ant1 |
⊢ ( ( 𝐴 ∈ No ∧ 𝑋 ∈ On ∧ ( 𝐴 ‘ 𝑋 ) = ∅ ) → Fun 𝐴 ) |
9 |
|
fvelrn |
⊢ ( ( Fun 𝐴 ∧ 𝑋 ∈ dom 𝐴 ) → ( 𝐴 ‘ 𝑋 ) ∈ ran 𝐴 ) |
10 |
8 9
|
sylan |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝑋 ∈ On ∧ ( 𝐴 ‘ 𝑋 ) = ∅ ) ∧ 𝑋 ∈ dom 𝐴 ) → ( 𝐴 ‘ 𝑋 ) ∈ ran 𝐴 ) |
11 |
6 10
|
sseldd |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝑋 ∈ On ∧ ( 𝐴 ‘ 𝑋 ) = ∅ ) ∧ 𝑋 ∈ dom 𝐴 ) → ( 𝐴 ‘ 𝑋 ) ∈ { 1o , 2o } ) |
12 |
3 11
|
eqeltrrd |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝑋 ∈ On ∧ ( 𝐴 ‘ 𝑋 ) = ∅ ) ∧ 𝑋 ∈ dom 𝐴 ) → ∅ ∈ { 1o , 2o } ) |
13 |
2 12
|
mtand |
⊢ ( ( 𝐴 ∈ No ∧ 𝑋 ∈ On ∧ ( 𝐴 ‘ 𝑋 ) = ∅ ) → ¬ 𝑋 ∈ dom 𝐴 ) |
14 |
|
nodmon |
⊢ ( 𝐴 ∈ No → dom 𝐴 ∈ On ) |
15 |
14
|
3ad2ant1 |
⊢ ( ( 𝐴 ∈ No ∧ 𝑋 ∈ On ∧ ( 𝐴 ‘ 𝑋 ) = ∅ ) → dom 𝐴 ∈ On ) |
16 |
|
simp2 |
⊢ ( ( 𝐴 ∈ No ∧ 𝑋 ∈ On ∧ ( 𝐴 ‘ 𝑋 ) = ∅ ) → 𝑋 ∈ On ) |
17 |
|
ontri1 |
⊢ ( ( dom 𝐴 ∈ On ∧ 𝑋 ∈ On ) → ( dom 𝐴 ⊆ 𝑋 ↔ ¬ 𝑋 ∈ dom 𝐴 ) ) |
18 |
15 16 17
|
syl2anc |
⊢ ( ( 𝐴 ∈ No ∧ 𝑋 ∈ On ∧ ( 𝐴 ‘ 𝑋 ) = ∅ ) → ( dom 𝐴 ⊆ 𝑋 ↔ ¬ 𝑋 ∈ dom 𝐴 ) ) |
19 |
13 18
|
mpbird |
⊢ ( ( 𝐴 ∈ No ∧ 𝑋 ∈ On ∧ ( 𝐴 ‘ 𝑋 ) = ∅ ) → dom 𝐴 ⊆ 𝑋 ) |