Theorem uni0 3925

Description: The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Reproved without relying on ax-nul by Eric Schmidt.) (Contributed by NM, 16-Sep-1993.) (Revised by Eric Schmidt, 4-Apr-2007.)

Assertion

Ref	Expression
uni0	⊢ ∪ ∅ = ∅

Proof of Theorem uni0

Step	Hyp	Ref	Expression
1		0ss 3535	. 2 ⊢ ∅ ⊆ {∅}
2		uni0b 3923	. 2 ⊢ (∪ ∅ = ∅ ↔ ∅ ⊆ {∅})
3	1, 2	mpbir 146	1 ⊢ ∪ ∅ = ∅

Syntax hints:   = wceq 1398   ⊆ wss 3201  ∅c0 3496  {csn 3673  ∪ cuni 3898

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-dif 3203  df-in 3207  df-ss 3214  df-nul 3497  df-sn 3679  df-uni 3899

This theorem is referenced by:  iununir  4059  nnpredcl  4727  unixp0im  5280  iotanul  5309  1st0  6316  2nd0  6317  brtpos0  6461  tpostpos  6473  nnsucuniel  6706  sup00  7245  nnnninfeq2  7371  0opn  14797