Description: An ordinal greater than one is greater than zero. (Contributed by Mario Carneiro, 25-May-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | dif20el | ⊢ ( 𝐴 ∈ ( On ∖ 2o ) → ∅ ∈ 𝐴 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ondif2 | ⊢ ( 𝐴 ∈ ( On ∖ 2o ) ↔ ( 𝐴 ∈ On ∧ 1o ∈ 𝐴 ) ) | |
2 | 1 | simprbi | ⊢ ( 𝐴 ∈ ( On ∖ 2o ) → 1o ∈ 𝐴 ) |
3 | 0lt1o | ⊢ ∅ ∈ 1o | |
4 | eldifi | ⊢ ( 𝐴 ∈ ( On ∖ 2o ) → 𝐴 ∈ On ) | |
5 | ontr1 | ⊢ ( 𝐴 ∈ On → ( ( ∅ ∈ 1o ∧ 1o ∈ 𝐴 ) → ∅ ∈ 𝐴 ) ) | |
6 | 4 5 | syl | ⊢ ( 𝐴 ∈ ( On ∖ 2o ) → ( ( ∅ ∈ 1o ∧ 1o ∈ 𝐴 ) → ∅ ∈ 𝐴 ) ) |
7 | 3 6 | mpani | ⊢ ( 𝐴 ∈ ( On ∖ 2o ) → ( 1o ∈ 𝐴 → ∅ ∈ 𝐴 ) ) |
8 | 2 7 | mpd | ⊢ ( 𝐴 ∈ ( On ∖ 2o ) → ∅ ∈ 𝐴 ) |