Theorem untangtr 35921

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 5207	. . . 4 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
2		ssralv 4003	. . . 4 ⊢ (∪ 𝐴 ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥))
3	1, 2	sylbi 217	. . 3 ⊢ (Tr 𝐴 → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥))
4		elequ1 2121	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
5		elequ2 2129	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
6	4, 5	bitrd 279	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
7	6	notbid 318	. . . . 5 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦))
8	7	cbvralvw 3215	. . . 4 ⊢ (∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑦 ∈ ∪ 𝐴 ¬ 𝑦 ∈ 𝑦)
9		untuni 35916	. . . 4 ⊢ (∀𝑦 ∈ ∪ 𝐴 ¬ 𝑦 ∈ 𝑦 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦)
10	8, 9	bitri 275	. . 3 ⊢ (∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦)
11	3, 10	imbitrdi 251	. 2 ⊢ (Tr 𝐴 → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦))
12		untelirr 35915	. . 3 ⊢ (∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥)
13	12	ralimi 3074	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥)
14	11, 13	impbid1 225	1 ⊢ (Tr 𝐴 → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wral 3052   ⊆ wss 3902  ∪ cuni 4864  Tr wtr 5206

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062  df-v 3443  df-ss 3919  df-uni 4865  df-tr 5207

This theorem is referenced by: (None)