Theorem 0iun 3970

Description: An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
0iun	⊢ ∪ 𝑥 ∈ ∅ 𝐴 = ∅

Proof of Theorem 0iun

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rex0 3464	. . . 4 ⊢ ¬ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴
2		eliun 3916	. . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴 ↔ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴)
3	1, 2	mtbir 672	. . 3 ⊢ ¬ 𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴
4		noel 3450	. . 3 ⊢ ¬ 𝑦 ∈ ∅
5	3, 4	2false 702	. 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴 ↔ 𝑦 ∈ ∅)
6	5	eqriv 2190	1 ⊢ ∪ 𝑥 ∈ ∅ 𝐴 = ∅

Syntax hints:   = wceq 1364   ∈ wcel 2164  ∃wrex 2473  ∅c0 3446  ∪ 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-in1 615  ax-in2 616  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-fal 1370  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-dif 3155  df-nul 3447  df-iun 3914

This theorem is referenced by:  iununir  3996  rdg0  6440  iunfidisj  7005  fsum2d  11578  fsumiun  11620  fprod2d  11766  iuncld  14283