Theorem univ 5397

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 4842	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 4857	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 5396	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2765	1 ⊢ ∪ V = V

Syntax hints:   = wceq 1547  Vcvv 3432  𝒫 cpw 4536  ∪ cuni 4845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-un 3895  df-ss 3907  df-pw 4538  df-sn 4563  df-pr 4565  df-uni 4846

This theorem is referenced by: (None)