Theorem iun0 4023

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 3555	. . . . . 6 ⊢ ¬ y ∈ ∅
2	1	a1i 10	. . . . 5 ⊢ (x ∈ A → ¬ y ∈ ∅)
3	2	nrex 2717	. . . 4 ⊢ ¬ ∃x ∈ A y ∈ ∅
4		eliun 3974	. . . 4 ⊢ (y ∈ ∪x ∈ A ∅ ↔ ∃x ∈ A y ∈ ∅)
5	3, 4	mtbir 290	. . 3 ⊢ ¬ y ∈ ∪x ∈ A ∅
6	5, 1	2false 339	. 2 ⊢ (y ∈ ∪x ∈ A ∅ ↔ y ∈ ∅)
7	6	eqriv 2350	1 ⊢ ∪x ∈ A ∅ = ∅

Syntax hints:  ¬ wn 3   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  ∅c0 3551  ∪ciun 3970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552  df-iun 3972

This theorem is referenced by:  iununi  4051