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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalemclccjdd | Structured version Visualization version GIF version |
Description: Lemma for dath 38189. 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 |
---|---|
dalemclccjdd | ⊢ (𝜓 → 𝐶 ≤ (𝑐 ∨ 𝑑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | da.ps0 | . 2 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑)))) | |
2 | simp33 1211 | . 2 ⊢ (((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))) → 𝐶 ≤ (𝑐 ∨ 𝑑)) | |
3 | 1, 2 | sylbi 216 | 1 ⊢ (𝜓 → 𝐶 ≤ (𝑐 ∨ 𝑑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 ∈ wcel 2106 ≠ wne 2943 class class class wbr 5105 (class class class)co 7356 |
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 397 df-3an 1089 |
This theorem is referenced by: dalemswapyzps 38143 dalemrotps 38144 dalem21 38147 dalem25 38151 |
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