Theorem univ 4207

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 3579	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 3590	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 3953	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2062	1 ⊢ ∪ V = V

Syntax hints:   = wceq 1243  Vcvv 2557  𝒫 cpw 3359  ∪ cuni 3580

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-uni 3581

This theorem is referenced by: (None)