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Theorem ex-un 28207
Description: Example for df-un 3913. 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 4117 . . 3 (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ ({1} ∪ {8}))
2 snsspr1 4720 . . . . 5 {1} ⊆ {1, 3}
3 ssequn2 4134 . . . . 5 ({1} ⊆ {1, 3} ↔ ({1, 3} ∪ {1}) = {1, 3})
42, 3mpbi 233 . . . 4 ({1, 3} ∪ {1}) = {1, 3}
54uneq1i 4110 . . 3 (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ {8})
61, 5eqtr3i 2847 . 2 ({1, 3} ∪ ({1} ∪ {8})) = ({1, 3} ∪ {8})
7 df-pr 4542 . . 3 {1, 8} = ({1} ∪ {8})
87uneq2i 4111 . 2 ({1, 3} ∪ {1, 8}) = ({1, 3} ∪ ({1} ∪ {8}))
9 df-tp 4544 . 2 {1, 3, 8} = ({1, 3} ∪ {8})
106, 8, 93eqtr4i 2855 1 ({1, 3} ∪ {1, 8}) = {1, 3, 8}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  cun 3906  wss 3908  {csn 4539  {cpr 4541  {ctp 4543  1c1 10527  3c3 11681  8c8 11686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-v 3471  df-un 3913  df-in 3915  df-ss 3925  df-pr 4542  df-tp 4544
This theorem is referenced by:  ex-uni  28209
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