Theorem ex-un 30172

Description: Example for df-un 3946. 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 4159	. . 3 ⊢ (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ ({1} ∪ {8}))
2		snsspr1 4810	. . . . 5 ⊢ {1} ⊆ {1, 3}
3		ssequn2 4176	. . . . 5 ⊢ ({1} ⊆ {1, 3} ↔ ({1, 3} ∪ {1}) = {1, 3})
4	2, 3	mpbi 229	. . . 4 ⊢ ({1, 3} ∪ {1}) = {1, 3}
5	4	uneq1i 4152	. . 3 ⊢ (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ {8})
6	1, 5	eqtr3i 2754	. 2 ⊢ ({1, 3} ∪ ({1} ∪ {8})) = ({1, 3} ∪ {8})
7		df-pr 4624	. . 3 ⊢ {1, 8} = ({1} ∪ {8})
8	7	uneq2i 4153	. 2 ⊢ ({1, 3} ∪ {1, 8}) = ({1, 3} ∪ ({1} ∪ {8}))
9		df-tp 4626	. 2 ⊢ {1, 3, 8} = ({1, 3} ∪ {8})
10	6, 8, 9	3eqtr4i 2762	1 ⊢ ({1, 3} ∪ {1, 8}) = {1, 3, 8}

Syntax hints:   = wceq 1533   ∪ cun 3939   ⊆ wss 3941  {csn 4621  {cpr 4623  {ctp 4625  1c1 11108  3c3 12267  8c8 12272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-un 3946  df-in 3948  df-ss 3958  df-pr 4624  df-tp 4626

This theorem is referenced by:  ex-uni  30174