Theorem uni0 3851

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 3476	. 2 ⊢ ∅ ⊆ {∅}
2		uni0b 3849	. 2 ⊢ (∪ ∅ = ∅ ↔ ∅ ⊆ {∅})
3	1, 2	mpbir 146	1 ⊢ ∪ ∅ = ∅

Syntax hints:   = wceq 1364   ⊆ wss 3144  ∅c0 3437  {csn 3607  ∪ cuni 3824

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157  df-nul 3438  df-sn 3613  df-uni 3825

This theorem is referenced by:  iununir  3985  nnpredcl  4640  unixp0im  5183  iotanul  5211  1st0  6169  2nd0  6170  brtpos0  6277  tpostpos  6289  nnsucuniel  6520  sup00  7032  nnnninfeq2  7157  0opn  13963