Theorem 0iun 3985

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 3478	. . . 4 ⊢ ¬ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴
2		eliun 3931	. . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴 ↔ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴)
3	1, 2	mtbir 673	. . 3 ⊢ ¬ 𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴
4		noel 3464	. . 3 ⊢ ¬ 𝑦 ∈ ∅
5	3, 4	2false 703	. 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴 ↔ 𝑦 ∈ ∅)
6	5	eqriv 2202	1 ⊢ ∪ 𝑥 ∈ ∅ 𝐴 = ∅

Syntax hints:   = wceq 1373   ∈ wcel 2176  ∃wrex 2485  ∅c0 3460  ∪ ciun 3927

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-dif 3168  df-nul 3461  df-iun 3929

This theorem is referenced by:  iununir  4011  rdg0  6473  iunfidisj  7048  fsum2d  11746  fsumiun  11788  fprod2d  11934  iuncld  14587