Theorem uni0 3867

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 3490	. 2 ⊢ ∅ ⊆ {∅}
2		uni0b 3865	. 2 ⊢ (∪ ∅ = ∅ ↔ ∅ ⊆ {∅})
3	1, 2	mpbir 146	1 ⊢ ∪ ∅ = ∅

Syntax hints:   = wceq 1364   ⊆ wss 3157  ∅c0 3451  {csn 3623  ∪ cuni 3840

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-in 3163  df-ss 3170  df-nul 3452  df-sn 3629  df-uni 3841

This theorem is referenced by:  iununir  4001  nnpredcl  4660  unixp0im  5207  iotanul  5235  1st0  6211  2nd0  6212  brtpos0  6319  tpostpos  6331  nnsucuniel  6562  sup00  7078  nnnninfeq2  7204  0opn  14326