Step |
Hyp |
Ref |
Expression |
1 |
|
frege130.m |
⊢ 𝑀 ∈ 𝑈 |
2 |
|
frege130.r |
⊢ 𝑅 ∈ 𝑉 |
3 |
1 2
|
frege131 |
⊢ ( Fun ◡ ◡ 𝑅 → 𝑅 hereditary ( ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) ) |
4 |
|
frege9 |
⊢ ( ( Fun ◡ ◡ 𝑅 → 𝑅 hereditary ( ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) ) → ( ( 𝑅 hereditary ( ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑀 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) ) ) ) → ( Fun ◡ ◡ 𝑅 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑀 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) ) ) ) ) ) |
5 |
3 4
|
ax-mp |
⊢ ( ( 𝑅 hereditary ( ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑀 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) ) ) ) → ( Fun ◡ ◡ 𝑅 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑀 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) ) ) ) ) |