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Theorem ex-un 30306
Description: Example for df-un 3949. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
Assertion
Ref Expression
ex-un ({1, 3} ∪ {1, 8}) = {1, 3, 8}

Proof of Theorem ex-un
StepHypRef Expression
1 unass 4164 . . 3 (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ ({1} ∪ {8}))
2 snsspr1 4819 . . . . 5 {1} ⊆ {1, 3}
3 ssequn2 4181 . . . . 5 ({1} ⊆ {1, 3} ↔ ({1, 3} ∪ {1}) = {1, 3})
42, 3mpbi 229 . . . 4 ({1, 3} ∪ {1}) = {1, 3}
54uneq1i 4156 . . 3 (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ {8})
61, 5eqtr3i 2755 . 2 ({1, 3} ∪ ({1} ∪ {8})) = ({1, 3} ∪ {8})
7 df-pr 4633 . . 3 {1, 8} = ({1} ∪ {8})
87uneq2i 4157 . 2 ({1, 3} ∪ {1, 8}) = ({1, 3} ∪ ({1} ∪ {8}))
9 df-tp 4635 . 2 {1, 3, 8} = ({1, 3} ∪ {8})
106, 8, 93eqtr4i 2763 1 ({1, 3} ∪ {1, 8}) = {1, 3, 8}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  cun 3942  wss 3944  {csn 4630  {cpr 4632  {ctp 4634  1c1 11141  3c3 12301  8c8 12306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3463  df-un 3949  df-ss 3961  df-pr 4633  df-tp 4635
This theorem is referenced by:  ex-uni  30308
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