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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem-ddly | Structured version Visualization version GIF version |
Description: Lemma for dath 38607. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.) |
Ref | Expression |
---|---|
da.ps0 | ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑)))) |
Ref | Expression |
---|---|
dalem-ddly | ⊢ (𝜓 → ¬ 𝑑 ≤ 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | da.ps0 | . 2 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑)))) | |
2 | simp32 1211 | . 2 ⊢ (((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))) → ¬ 𝑑 ≤ 𝑌) | |
3 | 1, 2 | sylbi 216 | 1 ⊢ (𝜓 → ¬ 𝑑 ≤ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 ∈ wcel 2107 ≠ wne 2941 class class class wbr 5149 (class class class)co 7409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 206 df-an 398 df-3an 1090 |
This theorem is referenced by: dalemswapyzps 38561 dalemrotps 38562 dalem23 38567 dalem24 38568 dalem25 38569 |
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