New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  uniexg GIF version

Theorem uniexg 4316
 Description: The sum class of a set is a set. (Contributed by SF, 14-Jan-2015.)
Assertion
Ref Expression
uniexg (A VA V)

Proof of Theorem uniexg
StepHypRef Expression
1 dfuni3 4315 . 2 A = ⋃1(k Skk A)
2 ssetkex 4294 . . . . 5 Sk V
32cnvkex 4287 . . . 4 k Sk V
4 imakexg 4299 . . . 4 ((k Sk V A V) → (k Skk A) V)
53, 4mpan 651 . . 3 (A V → (k Skk A) V)
6 uni1exg 4292 . . 3 ((k Skk A) V → ⋃1(k Skk A) V)
75, 6syl 15 . 2 (A V → ⋃1(k Skk A) V)
81, 7syl5eqel 2437 1 (A VA V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1710  Vcvv 2859  ∪cuni 3891  ⋃1cuni1 4133  ◡kccnvk 4175   “k cimak 4179   Sk cssetk 4183 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-uni 3892  df-opk 4058  df-1c 4136  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-imak 4189  df-p6 4191  df-sik 4192  df-ssetk 4193 This theorem is referenced by:  uniex  4317  pw1exb  4326
 Copyright terms: Public domain W3C validator