Theorem untuni 30840

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 3005	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥) ↔ (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥))
2	1	albii 1737	. . 3 ⊢ (∀𝑥∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥) ↔ ∀𝑥(∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥))
3		ralcom4 3197	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥(𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥) ↔ ∀𝑥∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥))
4		eluni2 4376	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
5	4	imbi1i 338	. . . 4 ⊢ ((𝑥 ∈ ∪ 𝐴 → ¬ 𝑥 ∈ 𝑥) ↔ (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥))
6	5	albii 1737	. . 3 ⊢ (∀𝑥(𝑥 ∈ ∪ 𝐴 → ¬ 𝑥 ∈ 𝑥) ↔ ∀𝑥(∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥))
7	2, 3, 6	3bitr4ri 292	. 2 ⊢ (∀𝑥(𝑥 ∈ ∪ 𝐴 → ¬ 𝑥 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝐴 ∀𝑥(𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥))
8		df-ral 2901	. 2 ⊢ (∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑥(𝑥 ∈ ∪ 𝐴 → ¬ 𝑥 ∈ 𝑥))
9		df-ral 2901	. . 3 ⊢ (∀𝑥 ∈ 𝑦 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑥(𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥))
10	9	ralbii 2963	. 2 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝑦 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑦 ∈ 𝐴 ∀𝑥(𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑥))
11	7, 8, 10	3bitr4i 291	1 ⊢ (∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝑦 ¬ 𝑥 ∈ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195  ∀wal 1473   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  ∪ cuni 4372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-uni 4373

This theorem is referenced by:  untangtr  30845  dfon2lem3  30934  dfon2lem7  30938