Theorem uni0 3838

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 3463	. 2 ⊢ ∅ ⊆ {∅}
2		uni0b 3836	. 2 ⊢ (∪ ∅ = ∅ ↔ ∅ ⊆ {∅})
3	1, 2	mpbir 146	1 ⊢ ∪ ∅ = ∅

Syntax hints:   = wceq 1353   ⊆ wss 3131  ∅c0 3424  {csn 3594  ∪ cuni 3811

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-dif 3133  df-in 3137  df-ss 3144  df-nul 3425  df-sn 3600  df-uni 3812

This theorem is referenced by:  iununir  3972  nnpredcl  4624  unixp0im  5167  iotanul  5195  1st0  6147  2nd0  6148  brtpos0  6255  tpostpos  6267  nnsucuniel  6498  sup00  7004  nnnninfeq2  7129  0opn  13591