Theorem ex-un 30572

Description: Example for df-un 3909. 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

Step	Hyp	Ref	Expression
1		unass 4124	. . 3 ⊢ (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ ({1} ∪ {8}))
2		snsspr1 4771	. . . . 5 ⊢ {1} ⊆ {1, 3}
3		ssequn2 4141	. . . . 5 ⊢ ({1} ⊆ {1, 3} ↔ ({1, 3} ∪ {1}) = {1, 3})
4	2, 3	mpbi 232	. . . 4 ⊢ ({1, 3} ∪ {1}) = {1, 3}
5	4	uneq1i 4117	. . 3 ⊢ (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ {8})
6	1, 5	eqtr3i 2786	. 2 ⊢ ({1, 3} ∪ ({1} ∪ {8})) = ({1, 3} ∪ {8})
7		df-pr 4584	. . 3 ⊢ {1, 8} = ({1} ∪ {8})
8	7	uneq2i 4118	. 2 ⊢ ({1, 3} ∪ {1, 8}) = ({1, 3} ∪ ({1} ∪ {8}))
9		df-tp 4586	. 2 ⊢ {1, 3, 8} = ({1, 3} ∪ {8})
10	6, 8, 9	3eqtr4i 2794	1 ⊢ ({1, 3} ∪ {1, 8}) = {1, 3, 8}

Syntax hints:   = wceq 1559   ∪ cun 3902   ⊆ wss 3904  {csn 4581  {cpr 4583  {ctp 4585  1c1 11071  3c3 12270  8c8 12275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-un 3909  df-ss 3921  df-pr 4584  df-tp 4586

This theorem is referenced by:  ex-uni  30574