Theorem untangtr 32040

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 4914	. . . 4 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
2		ssralv 3828	. . . 4 ⊢ (∪ 𝐴 ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥))
3	1, 2	sylbi 208	. . 3 ⊢ (Tr 𝐴 → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥))
4		elequ1 2162	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
5		elequ2 2169	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
6	4, 5	bitrd 270	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
7	6	notbid 309	. . . . 5 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦))
8	7	cbvralv 3319	. . . 4 ⊢ (∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑦 ∈ ∪ 𝐴 ¬ 𝑦 ∈ 𝑦)
9		untuni 32035	. . . 4 ⊢ (∀𝑦 ∈ ∪ 𝐴 ¬ 𝑦 ∈ 𝑦 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦)
10	8, 9	bitri 266	. . 3 ⊢ (∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦)
11	3, 10	syl6ib 242	. 2 ⊢ (Tr 𝐴 → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦))
12		untelirr 32034	. . 3 ⊢ (∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥)
13	12	ralimi 3099	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥)
14	11, 13	impbid1 216	1 ⊢ (Tr 𝐴 → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 197  ∀wral 3055   ⊆ wss 3734  ∪ cuni 4596  Tr wtr 4913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-v 3352  df-in 3741  df-ss 3748  df-uni 4597  df-tr 4914

This theorem is referenced by: (None)