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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalemccea | Structured version Visualization version GIF version |
Description: Lemma for dath 37747. 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 |
---|---|
dalemccea | ⊢ (𝜓 → 𝑐 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | da.ps0 | . 2 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑)))) | |
2 | simp1l 1196 | . 2 ⊢ (((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))) → 𝑐 ∈ 𝐴) | |
3 | 1, 2 | sylbi 216 | 1 ⊢ (𝜓 → 𝑐 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 ∈ wcel 2106 ≠ wne 2943 class class class wbr 5076 (class class class)co 7277 |
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 1088 |
This theorem is referenced by: dalemcceb 37700 dalemswapyzps 37701 dalemrotps 37702 dalemcjden 37703 dalem23 37707 dalem24 37708 dalem25 37709 dalem27 37710 dalem28 37711 dalem38 37721 dalem39 37722 dalem44 37727 dalem51 37734 dalem56 37739 |
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