Theorem iuncom 4963

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 3293	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐶)
2		eliun 4959	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ 𝐵 𝐶 ↔ ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶)
3	2	rexbii 3111	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ ∪ 𝑦 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶)
4		eliun 4959	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐶)
5	4	rexbii 3111	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐶)
6	1, 3, 5	3bitr4i 306	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ ∪ 𝑦 ∈ 𝐵 𝐶 ↔ ∃𝑦 ∈ 𝐵 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶)
7		eliun 4959	. . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ∪ 𝑦 ∈ 𝐵 𝐶)
8		eliun 4959	. . 3 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑦 ∈ 𝐵 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶)
9	6, 7, 8	3bitr4i 306	. 2 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 ↔ 𝑧 ∈ ∪ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶)
10	9	eqriv 2759	1 ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 = ∪ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2142  ∃wrex 3088  ∪ ciun 4955

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-11 2191  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-v 3456  df-iun 4957

This theorem is used by:  pzriprnglem11  21652