Theorem 0iun 4065

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 3539	. . . 4 ⊢ ¬ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴
2		eliun 4011	. . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴 ↔ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴)
3	1, 2	mtbir 682	. . 3 ⊢ ¬ 𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴
4		noel 3525	. . 3 ⊢ ¬ 𝑦 ∈ ∅
5	3, 4	2false 713	. 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴 ↔ 𝑦 ∈ ∅)
6	5	eqriv 2235	1 ⊢ ∪ 𝑥 ∈ ∅ 𝐴 = ∅

Syntax hints:   = wceq 1402   ∈ wcel 2209  ∃wrex 2529  ∅c0 3520  ∪ ciun 4007

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-nul 3521  df-iun 4009

This theorem is referenced by:  iununir  4091  rdg0  6648  iunfidisj  7250  fsum2d  12180  fsumiun  12222  fprod2d  12368  iuncld  15139