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Theorem ex-un 27169
Description: Example for df-un 3565. 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 3754 . . 3 (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ ({1} ∪ {8}))
2 snsspr1 4320 . . . . 5 {1} ⊆ {1, 3}
3 ssequn2 3770 . . . . 5 ({1} ⊆ {1, 3} ↔ ({1, 3} ∪ {1}) = {1, 3})
42, 3mpbi 220 . . . 4 ({1, 3} ∪ {1}) = {1, 3}
54uneq1i 3747 . . 3 (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ {8})
61, 5eqtr3i 2645 . 2 ({1, 3} ∪ ({1} ∪ {8})) = ({1, 3} ∪ {8})
7 df-pr 4158 . . 3 {1, 8} = ({1} ∪ {8})
87uneq2i 3748 . 2 ({1, 3} ∪ {1, 8}) = ({1, 3} ∪ ({1} ∪ {8}))
9 df-tp 4160 . 2 {1, 3, 8} = ({1, 3} ∪ {8})
106, 8, 93eqtr4i 2653 1 ({1, 3} ∪ {1, 8}) = {1, 3, 8}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  cun 3558  wss 3560  {csn 4155  {cpr 4157  {ctp 4159  1c1 9897  3c3 11031  8c8 11036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3192  df-un 3565  df-in 3567  df-ss 3574  df-pr 4158  df-tp 4160
This theorem is referenced by:  ex-uni  27171
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