Theorem iuncom 4955

Description: Commutation of indexed unions. (Contributed by NM, 18-Dec-2008.)

Assertion

Ref	Expression
iuncom	⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 = ∪ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶

Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem iuncom

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexcom 3266	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐶)
2		eliun 4951	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ 𝐵 𝐶 ↔ ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶)
3	2	rexbii 3084	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ ∪ 𝑦 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶)
4		eliun 4951	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐶)
5	4	rexbii 3084	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐶)
6	1, 3, 5	3bitr4i 303	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ ∪ 𝑦 ∈ 𝐵 𝐶 ↔ ∃𝑦 ∈ 𝐵 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶)
7		eliun 4951	. . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ∪ 𝑦 ∈ 𝐵 𝐶)
8		eliun 4951	. . 3 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑦 ∈ 𝐵 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶)
9	6, 7, 8	3bitr4i 303	. 2 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 ↔ 𝑧 ∈ ∪ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶)
10	9	eqriv 2734	1 ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 = ∪ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶

Syntax hints:   = wceq 1542   ∈ wcel 2114  ∃wrex 3061  ∪ ciun 4947

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062  df-v 3443  df-iun 4949

This theorem is referenced by:  pzriprnglem11  21450