Theorem untangtr 34984

Description: A transitive class is untangled iff its elements are. (Contributed by Scott Fenton, 7-Mar-2011.)

Assertion

Ref	Expression
untangtr	⊢ (Tr 𝐴 → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦))

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem untangtr

Step	Hyp	Ref	Expression
1		df-tr 5267	. . . 4 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
2		ssralv 4051	. . . 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 2112	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
5		elequ2 2120	. . . . . . 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 3233	. . . 4 ⊢ (∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑦 ∈ ∪ 𝐴 ¬ 𝑦 ∈ 𝑦)
9		untuni 34979	. . . 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 34978	. . 3 ⊢ (∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥)
13	12	ralimi 3082	. 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))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205  ∀wral 3060   ⊆ wss 3949  ∪ cuni 4909  Tr wtr 5266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-11 2153  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-v 3475  df-in 3956  df-ss 3966  df-uni 4910  df-tr 5267

This theorem is referenced by: (None)