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Theorem ex-un 28130
Description: Example for df-un 3938. 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 4139 . . 3 (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ ({1} ∪ {8}))
2 snsspr1 4739 . . . . 5 {1} ⊆ {1, 3}
3 ssequn2 4156 . . . . 5 ({1} ⊆ {1, 3} ↔ ({1, 3} ∪ {1}) = {1, 3})
42, 3mpbi 231 . . . 4 ({1, 3} ∪ {1}) = {1, 3}
54uneq1i 4132 . . 3 (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ {8})
61, 5eqtr3i 2843 . 2 ({1, 3} ∪ ({1} ∪ {8})) = ({1, 3} ∪ {8})
7 df-pr 4560 . . 3 {1, 8} = ({1} ∪ {8})
87uneq2i 4133 . 2 ({1, 3} ∪ {1, 8}) = ({1, 3} ∪ ({1} ∪ {8}))
9 df-tp 4562 . 2 {1, 3, 8} = ({1, 3} ∪ {8})
106, 8, 93eqtr4i 2851 1 ({1, 3} ∪ {1, 8}) = {1, 3, 8}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  cun 3931  wss 3933  {csn 4557  {cpr 4559  {ctp 4561  1c1 10526  3c3 11681  8c8 11686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-un 3938  df-in 3940  df-ss 3949  df-pr 4560  df-tp 4562
This theorem is referenced by:  ex-uni  28132
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