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Theorem ex-un 27984
 Description: Example for df-un 3836. 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 4033 . . 3 (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ ({1} ∪ {8}))
2 snsspr1 4622 . . . . 5 {1} ⊆ {1, 3}
3 ssequn2 4049 . . . . 5 ({1} ⊆ {1, 3} ↔ ({1, 3} ∪ {1}) = {1, 3})
42, 3mpbi 222 . . . 4 ({1, 3} ∪ {1}) = {1, 3}
54uneq1i 4026 . . 3 (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ {8})
61, 5eqtr3i 2804 . 2 ({1, 3} ∪ ({1} ∪ {8})) = ({1, 3} ∪ {8})
7 df-pr 4445 . . 3 {1, 8} = ({1} ∪ {8})
87uneq2i 4027 . 2 ({1, 3} ∪ {1, 8}) = ({1, 3} ∪ ({1} ∪ {8}))
9 df-tp 4447 . 2 {1, 3, 8} = ({1, 3} ∪ {8})
106, 8, 93eqtr4i 2812 1 ({1, 3} ∪ {1, 8}) = {1, 3, 8}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1507   ∪ cun 3829   ⊆ wss 3831  {csn 4442  {cpr 4444  {ctp 4446  1c1 10338  3c3 11499  8c8 11504 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-v 3417  df-un 3836  df-in 3838  df-ss 3845  df-pr 4445  df-tp 4447 This theorem is referenced by:  ex-uni  27986
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