Theorem univ 5394

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 4855	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 4870	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 5393	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2758	1 ⊢ ∪ V = V

Syntax hints:   = wceq 1541  Vcvv 3437  𝒫 cpw 4549  ∪ cuni 4858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-un 3903  df-ss 3915  df-pw 4551  df-sn 4576  df-pr 4578  df-uni 4859

This theorem is referenced by: (None)