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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalemccnedd | Structured version Visualization version GIF version |
Description: Lemma for dath 35543. 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 |
---|---|
dalemccnedd | ⊢ (𝜓 → 𝑐 ≠ 𝑑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | da.ps0 | . . 3 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑)))) | |
2 | simp31 1252 | . . 3 ⊢ (((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))) → 𝑑 ≠ 𝑐) | |
3 | 1, 2 | sylbi 207 | . 2 ⊢ (𝜓 → 𝑑 ≠ 𝑐) |
4 | 3 | necomd 2987 | 1 ⊢ (𝜓 → 𝑐 ≠ 𝑑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1072 ∈ wcel 2139 ≠ wne 2932 class class class wbr 4804 (class class class)co 6814 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-an 385 df-3an 1074 df-ex 1854 df-cleq 2753 df-ne 2933 |
This theorem is referenced by: dalemswapyzps 35497 dalemrotps 35498 dalemcjden 35499 |
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