Theorem univ 5109

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 4627	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 4639	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 5108	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2830	1 ⊢ ∪ V = V

Syntax hints:   = wceq 1637  Vcvv 3391  𝒫 cpw 4351  ∪ cuni 4630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-rex 3102  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-pw 4353  df-sn 4371  df-pr 4373  df-uni 4631

This theorem is referenced by: (None)