Theorem iun0 3922

Description: An indexed union of the empty set is empty. (Contributed by NM, 26-Mar-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
iun0	⊢ ∪ 𝑥 ∈ 𝐴 ∅ = ∅

Proof of Theorem iun0

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		noel 3413	. . . . . 6 ⊢ ¬ 𝑦 ∈ ∅
2	1	a1i 9	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ¬ 𝑦 ∈ ∅)
3	2	nrex 2558	. . . 4 ⊢ ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ ∅
4		eliun 3870	. . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∅ ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ∅)
5	3, 4	mtbir 661	. . 3 ⊢ ¬ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∅
6	5, 1	2false 691	. 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∅ ↔ 𝑦 ∈ ∅)
7	6	eqriv 2162	1 ⊢ ∪ 𝑥 ∈ 𝐴 ∅ = ∅

Syntax hints:  ¬ wn 3   = wceq 1343   ∈ wcel 2136  ∃wrex 2445  ∅c0 3409  ∪ ciun 3866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-nul 3410  df-iun 3868

This theorem is referenced by: (None)