Theorem untuni 33048

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 3238	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥) ↔ (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥))
2	1	albii 1821	. . 3 ⊢ (∀𝑥∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥) ↔ ∀𝑥(∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥))
3		ralcom4 3198	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥(𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥) ↔ ∀𝑥∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥))
4		eluni2 4804	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
5	4	imbi1i 353	. . . 4 ⊢ ((𝑥 ∈ ∪ 𝐴 → ¬ 𝑥 ∈ 𝑥) ↔ (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥))
6	5	albii 1821	. . 3 ⊢ (∀𝑥(𝑥 ∈ ∪ 𝐴 → ¬ 𝑥 ∈ 𝑥) ↔ ∀𝑥(∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥))
7	2, 3, 6	3bitr4ri 307	. 2 ⊢ (∀𝑥(𝑥 ∈ ∪ 𝐴 → ¬ 𝑥 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝐴 ∀𝑥(𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥))
8		df-ral 3111	. 2 ⊢ (∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑥(𝑥 ∈ ∪ 𝐴 → ¬ 𝑥 ∈ 𝑥))
9		df-ral 3111	. . 3 ⊢ (∀𝑥 ∈ 𝑦 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑥(𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥))
10	9	ralbii 3133	. 2 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝑦 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑦 ∈ 𝐴 ∀𝑥(𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥))
11	7, 8, 10	3bitr4i 306	1 ⊢ (∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝑦 ¬ 𝑥 ∈ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1536   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  ∪ cuni 4800

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-11 2158  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rex 3112  df-v 3443  df-uni 4801

This theorem is referenced by:  untangtr  33053  dfon2lem3  33143  dfon2lem7  33147