Theorem uni0 1938

Description: The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54.

Assertion

Ref	Expression
uni0	⊢ ∪∅ = ∅

Proof of Theorem uni0

Step	Hyp	Ref	Expression
1		0ss 1725	. . . 4 ⊢ ∅ ⊆ {∅}
2		uniss 1936	. . . 4 ⊢ (∅ ⊆ {∅} → ∪∅ ⊆ ∪{∅})
3	1, 2	ax-mp 6	. . 3 ⊢ ∪∅ ⊆ ∪{∅}
4		0ex 1745	. . . 4 ⊢ ∅ ∈ V
5	4	unisn 1932	. . 3 ⊢ ∪{∅} = ∅
6	3, 5	sseqtr 1532	. 2 ⊢ ∪∅ ⊆ ∅
7		ss0 1727	. 2 ⊢ (∪∅ ⊆ ∅ → ∪∅ = ∅)
8	6, 7	ax-mp 6	1 ⊢ ∪∅ = ∅

Syntax hints:   = wceq 1091   ⊆ wss 1487  ∅c0 1707  {csn 1808  ∪cuni 1919

This theorem is referenced by:  unizlim 2364  fvprc 2829  funfv 2862  fvopabn 2873  1stval 3089  2ndval 3090  1st2val 3097  inf5 3472

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-sn 1811  df-pr 1812  df-uni 1920

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