Theorem untuni 36023

Description: The union of a class is untangled iff all its members are untangled. (Contributed by Scott Fenton, 28-Feb-2011.)

Assertion

Ref	Expression
untuni	⊢ (∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝑦 ¬ 𝑥 ∈ 𝑥)

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem untuni

Step	Hyp	Ref	Expression
1		r19.23v 3188	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥) ↔ (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥))
2	1	albii 1838	. . 3 ⊢ (∀𝑥∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥) ↔ ∀𝑥(∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥))
3		ralcom4 3287	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥(𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥) ↔ ∀𝑥∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥))
4		eluni2 4868	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
5	4	imbi1i 351	. . . 4 ⊢ ((𝑥 ∈ ∪ 𝐴 → ¬ 𝑥 ∈ 𝑥) ↔ (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥))
6	5	albii 1838	. . 3 ⊢ (∀𝑥(𝑥 ∈ ∪ 𝐴 → ¬ 𝑥 ∈ 𝑥) ↔ ∀𝑥(∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥))
7	2, 3, 6	3bitr4ri 306	. 2 ⊢ (∀𝑥(𝑥 ∈ ∪ 𝐴 → ¬ 𝑥 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝐴 ∀𝑥(𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥))
8		df-ral 3076	. 2 ⊢ (∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑥(𝑥 ∈ ∪ 𝐴 → ¬ 𝑥 ∈ 𝑥))
9		df-ral 3076	. . 3 ⊢ (∀𝑥 ∈ 𝑦 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑥(𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥))
10	9	ralbii 3107	. 2 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝑦 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑦 ∈ 𝐴 ∀𝑥(𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥))
11	7, 8, 10	3bitr4i 305	1 ⊢ (∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝑦 ¬ 𝑥 ∈ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208  ∀wal 1557   ∈ wcel 2141  ∀wral 3075  ∃wrex 3085  ∪ cuni 4864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-11 2190  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-v 3455  df-uni 4865

This theorem is referenced by:  untangtr  36028  dfon2lem3  36097  dfon2lem7  36101