Theorem univ 5414

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 4871	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 4886	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 5413	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2755	1 ⊢ ∪ V = V

Syntax hints:   = wceq 1540  Vcvv 3450  𝒫 cpw 4566  ∪ cuni 4874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-un 3922  df-ss 3934  df-pw 4568  df-sn 4593  df-pr 4595  df-uni 4875

This theorem is referenced by: (None)