Theorem ex-un 30511

Description: Example for df-un 3908. 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 4126	. . 3 ⊢ (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ ({1} ∪ {8}))
2		snsspr1 4772	. . . . 5 ⊢ {1} ⊆ {1, 3}
3		ssequn2 4143	. . . . 5 ⊢ ({1} ⊆ {1, 3} ↔ ({1, 3} ∪ {1}) = {1, 3})
4	2, 3	mpbi 230	. . . 4 ⊢ ({1, 3} ∪ {1}) = {1, 3}
5	4	uneq1i 4118	. . 3 ⊢ (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ {8})
6	1, 5	eqtr3i 2762	. 2 ⊢ ({1, 3} ∪ ({1} ∪ {8})) = ({1, 3} ∪ {8})
7		df-pr 4585	. . 3 ⊢ {1, 8} = ({1} ∪ {8})
8	7	uneq2i 4119	. 2 ⊢ ({1, 3} ∪ {1, 8}) = ({1, 3} ∪ ({1} ∪ {8}))
9		df-tp 4587	. 2 ⊢ {1, 3, 8} = ({1, 3} ∪ {8})
10	6, 8, 9	3eqtr4i 2770	1 ⊢ ({1, 3} ∪ {1, 8}) = {1, 3, 8}

Syntax hints:   = wceq 1542   ∪ cun 3901   ⊆ wss 3903  {csn 4582  {cpr 4584  {ctp 4586  1c1 11039  3c3 12213  8c8 12218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-un 3908  df-ss 3920  df-pr 4585  df-tp 4587

This theorem is referenced by:  ex-uni  30513