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Metamath Proof Explorer
Description: Property of weak dominance, forward direction only. (Contributed by Mario Carneiro, 5-May-2015)
|
|
Ref |
Expression |
|
Assertion |
brwdomi |
⊢ ( 𝑋 ≼* 𝑌 → ( 𝑋 = ∅ ∨ ∃ 𝑧 𝑧 : 𝑌 –onto→ 𝑋 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
relwdom |
⊢ Rel ≼* |
| 2 |
1
|
brrelex2i |
⊢ ( 𝑋 ≼* 𝑌 → 𝑌 ∈ V ) |
| 3 |
|
brwdom |
⊢ ( 𝑌 ∈ V → ( 𝑋 ≼* 𝑌 ↔ ( 𝑋 = ∅ ∨ ∃ 𝑧 𝑧 : 𝑌 –onto→ 𝑋 ) ) ) |
| 4 |
2 3
|
syl |
⊢ ( 𝑋 ≼* 𝑌 → ( 𝑋 ≼* 𝑌 ↔ ( 𝑋 = ∅ ∨ ∃ 𝑧 𝑧 : 𝑌 –onto→ 𝑋 ) ) ) |
| 5 |
4
|
ibi |
⊢ ( 𝑋 ≼* 𝑌 → ( 𝑋 = ∅ ∨ ∃ 𝑧 𝑧 : 𝑌 –onto→ 𝑋 ) ) |