Theorem iun0 4021

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 3495	. . . . . 6 ⊢ ¬ 𝑦 ∈ ∅
2	1	a1i 9	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ¬ 𝑦 ∈ ∅)
3	2	nrex 2622	. . . 4 ⊢ ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ ∅
4		eliun 3968	. . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∅ ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ∅)
5	3, 4	mtbir 675	. . 3 ⊢ ¬ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∅
6	5, 1	2false 706	. 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∅ ↔ 𝑦 ∈ ∅)
7	6	eqriv 2226	1 ⊢ ∪ 𝑥 ∈ 𝐴 ∅ = ∅

Syntax hints:  ¬ wn 3   = wceq 1395   ∈ wcel 2200  ∃wrex 2509  ∅c0 3491  ∪ ciun 3964

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-nul 3492  df-iun 3966

This theorem is referenced by: (None)