Theorem iuncom 3918

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 2658	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐶)
2		eliun 3916	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ 𝐵 𝐶 ↔ ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶)
3	2	rexbii 2501	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ ∪ 𝑦 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶)
4		eliun 3916	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐶)
5	4	rexbii 2501	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐶)
6	1, 3, 5	3bitr4i 212	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ ∪ 𝑦 ∈ 𝐵 𝐶 ↔ ∃𝑦 ∈ 𝐵 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶)
7		eliun 3916	. . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ∪ 𝑦 ∈ 𝐵 𝐶)
8		eliun 3916	. . 3 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑦 ∈ 𝐵 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶)
9	6, 7, 8	3bitr4i 212	. 2 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 ↔ 𝑧 ∈ ∪ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶)
10	9	eqriv 2190	1 ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 = ∪ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶

Syntax hints:   = wceq 1364   ∈ wcel 2164  ∃wrex 2473  ∪ ciun 3912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-iun 3914

This theorem is referenced by: (None)