Theorem untangtr 36295

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 5217	. . . 4 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
2		ssralv 4003	. . . 4 ⊢ (∪ 𝐴 ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥))
3	1, 2	sylbi 220	. . 3 ⊢ (Tr 𝐴 → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥))
4		elequ1 2152	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
5		elequ2 2160	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
6	4, 5	bitrd 282	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
7	6	notbid 321	. . . . 5 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦))
8	7	cbvralvw 3242	. . . 4 ⊢ (∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑦 ∈ ∪ 𝐴 ¬ 𝑦 ∈ 𝑦)
9		untuni 36290	. . . 4 ⊢ (∀𝑦 ∈ ∪ 𝐴 ¬ 𝑦 ∈ 𝑦 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦)
10	8, 9	bitri 278	. . 3 ⊢ (∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦)
11	3, 10	imbitrdi 254	. 2 ⊢ (Tr 𝐴 → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦))
12		untelirr 36289	. . 3 ⊢ (∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥)
13	12	ralimi 3101	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥)
14	11, 13	impbid1 228	1 ⊢ (Tr 𝐴 → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wral 3078   ⊆ wss 3902  ∪ cuni 4870  Tr wtr 5216

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-v 3455  df-ss 3919  df-uni 4871  df-tr 5217

This theorem is used by: (None)