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Mirrors > Home > MPE Home > Th. List > Mathboxes > untangtr | Structured version Visualization version GIF version |
Description: A transitive class is untangled iff its elements are. (Contributed by Scott Fenton, 7-Mar-2011.) |
Ref | Expression |
---|---|
untangtr | ⊢ (Tr 𝐴 → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tr 5266 | . . . 4 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴) | |
2 | ssralv 4050 | . . . 4 ⊢ (∪ 𝐴 ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥)) | |
3 | 1, 2 | sylbi 216 | . . 3 ⊢ (Tr 𝐴 → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥)) |
4 | elequ1 2113 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥)) | |
5 | elequ2 2121 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦)) | |
6 | 4, 5 | bitrd 278 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦)) |
7 | 6 | notbid 317 | . . . . 5 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦)) |
8 | 7 | cbvralvw 3234 | . . . 4 ⊢ (∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑦 ∈ ∪ 𝐴 ¬ 𝑦 ∈ 𝑦) |
9 | untuni 34747 | . . . 4 ⊢ (∀𝑦 ∈ ∪ 𝐴 ¬ 𝑦 ∈ 𝑦 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦) | |
10 | 8, 9 | bitri 274 | . . 3 ⊢ (∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦) |
11 | 3, 10 | imbitrdi 250 | . 2 ⊢ (Tr 𝐴 → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦)) |
12 | untelirr 34746 | . . 3 ⊢ (∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥) | |
13 | 12 | ralimi 3083 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥) |
14 | 11, 13 | impbid1 224 | 1 ⊢ (Tr 𝐴 → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∀wral 3061 ⊆ wss 3948 ∪ cuni 4908 Tr wtr 5265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-11 2154 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-v 3476 df-in 3955 df-ss 3965 df-uni 4909 df-tr 5266 |
This theorem is referenced by: (None) |
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