Step |
Hyp |
Ref |
Expression |
1 |
|
nnaord |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) → ( 𝐴 ∈ 𝐵 ↔ ( 𝐶 +o 𝐴 ) ∈ ( 𝐶 +o 𝐵 ) ) ) |
2 |
|
nnacom |
⊢ ( ( 𝐶 ∈ ω ∧ 𝐴 ∈ ω ) → ( 𝐶 +o 𝐴 ) = ( 𝐴 +o 𝐶 ) ) |
3 |
2
|
ancoms |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐶 ∈ ω ) → ( 𝐶 +o 𝐴 ) = ( 𝐴 +o 𝐶 ) ) |
4 |
3
|
3adant2 |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) → ( 𝐶 +o 𝐴 ) = ( 𝐴 +o 𝐶 ) ) |
5 |
|
nnacom |
⊢ ( ( 𝐶 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐶 +o 𝐵 ) = ( 𝐵 +o 𝐶 ) ) |
6 |
5
|
ancoms |
⊢ ( ( 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) → ( 𝐶 +o 𝐵 ) = ( 𝐵 +o 𝐶 ) ) |
7 |
6
|
3adant1 |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) → ( 𝐶 +o 𝐵 ) = ( 𝐵 +o 𝐶 ) ) |
8 |
4 7
|
eleq12d |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) → ( ( 𝐶 +o 𝐴 ) ∈ ( 𝐶 +o 𝐵 ) ↔ ( 𝐴 +o 𝐶 ) ∈ ( 𝐵 +o 𝐶 ) ) ) |
9 |
1 8
|
bitrd |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) → ( 𝐴 ∈ 𝐵 ↔ ( 𝐴 +o 𝐶 ) ∈ ( 𝐵 +o 𝐶 ) ) ) |