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Theorem dfuni3 4315
 Description: Alternate definition of class union for existence proof. (Contributed by SF, 14-Jan-2015.)
Assertion
Ref Expression
dfuni3 1k Sk k

Proof of Theorem dfuni3
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2862 . . . . . 6
2 snex 4111 . . . . . 6
31, 2opkelcnvk 4250 . . . . 5 k Sk Sk
4 vex 2862 . . . . . 6
54, 1elssetk 4270 . . . . 5 Sk
63, 5bitri 240 . . . 4 k Sk
76rexbii 2639 . . 3 k Sk
84eluni1 4173 . . . 4 1k Sk k k Sk k
92elimak 4259 . . . 4 k Sk k k Sk
108, 9bitri 240 . . 3 1k Sk k k Sk
11 eluni2 3895 . . 3
127, 10, 113bitr4ri 269 . 2 1k Sk k
1312eqriv 2350 1 1k Sk k
 Colors of variables: wff setvar class Syntax hints:   wceq 1642   wcel 1710  wrex 2615  csn 3737  cuni 3891  copk 4057  ⋃1cuni1 4133  kccnvk 4175  kcimak 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-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-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-cnvk 4186  df-imak 4189  df-ssetk 4193 This theorem is referenced by:  uniexg  4316
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