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Theorem ex-un 27740
Description: Example for df-un 3737. 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 3932 . . 3 (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ ({1} ∪ {8}))
2 snsspr1 4499 . . . . 5 {1} ⊆ {1, 3}
3 ssequn2 3948 . . . . 5 ({1} ⊆ {1, 3} ↔ ({1, 3} ∪ {1}) = {1, 3})
42, 3mpbi 221 . . . 4 ({1, 3} ∪ {1}) = {1, 3}
54uneq1i 3925 . . 3 (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ {8})
61, 5eqtr3i 2789 . 2 ({1, 3} ∪ ({1} ∪ {8})) = ({1, 3} ∪ {8})
7 df-pr 4337 . . 3 {1, 8} = ({1} ∪ {8})
87uneq2i 3926 . 2 ({1, 3} ∪ {1, 8}) = ({1, 3} ∪ ({1} ∪ {8}))
9 df-tp 4339 . 2 {1, 3, 8} = ({1, 3} ∪ {8})
106, 8, 93eqtr4i 2797 1 ({1, 3} ∪ {1, 8}) = {1, 3, 8}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1652  cun 3730  wss 3732  {csn 4334  {cpr 4336  {ctp 4338  1c1 10190  3c3 11328  8c8 11333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-un 3737  df-in 3739  df-ss 3746  df-pr 4337  df-tp 4339
This theorem is referenced by:  ex-uni  27742
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