Theorem uni0 3686

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 3325	. 2 ⊢ ∅ ⊆ {∅}
2		uni0b 3684	. 2 ⊢ (∪ ∅ = ∅ ↔ ∅ ⊆ {∅})
3	1, 2	mpbir 145	1 ⊢ ∪ ∅ = ∅

Syntax hints:   = wceq 1290   ⊆ wss 3000  ∅c0 3287  {csn 3450  ∪ cuni 3659

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-dif 3002  df-in 3006  df-ss 3013  df-nul 3288  df-sn 3456  df-uni 3660

This theorem is referenced by:  iununir  3818  unixp0im  4980  iotanul  5008  1st0  5929  2nd0  5930  brtpos0  6031  tpostpos  6043  nnsucuniel  6270  sup00  6752  0opn  11759  nnpredcl  12156