Theorem uni0 3919

Description: The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Reproved without relying on ax-nul in set.mm 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 3580	. 2 ⊢ ∅ ⊆ {∅}
2		uni0b 3917	. 2 ⊢ (∪∅ = ∅ ↔ ∅ ⊆ {∅})
3	1, 2	mpbir 200	1 ⊢ ∪∅ = ∅

Syntax hints:   = wceq 1642   ⊆ wss 3258  ∅c0 3551  {csn 3738  ∪cuni 3892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-uni 3893

This theorem is referenced by:  uniintsn  3964  iununi  4051  iotanul  4355  dfiota4  4373  funfv  5376