Theorem univ 5462

Description: The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)

Assertion

Ref	Expression
univ	⊢ ∪ V = V

Proof of Theorem univ

Step	Hyp	Ref	Expression
1		pwv 4909	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 4924	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 5461	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2765	1 ⊢ ∪ V = V

Syntax hints:   = wceq 1537  Vcvv 3478  𝒫 cpw 4605  ∪ cuni 4912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-ss 3980  df-pw 4607  df-sn 4632  df-pr 4634  df-uni 4913

This theorem is referenced by: (None)