Theorem ex-un 29075

Description: Example for df-un 3906. 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 4117	. . 3 ⊢ (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ ({1} ∪ {8}))
2		snsspr1 4765	. . . . 5 ⊢ {1} ⊆ {1, 3}
3		ssequn2 4134	. . . . 5 ⊢ ({1} ⊆ {1, 3} ↔ ({1, 3} ∪ {1}) = {1, 3})
4	2, 3	mpbi 229	. . . 4 ⊢ ({1, 3} ∪ {1}) = {1, 3}
5	4	uneq1i 4110	. . 3 ⊢ (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ {8})
6	1, 5	eqtr3i 2767	. 2 ⊢ ({1, 3} ∪ ({1} ∪ {8})) = ({1, 3} ∪ {8})
7		df-pr 4580	. . 3 ⊢ {1, 8} = ({1} ∪ {8})
8	7	uneq2i 4111	. 2 ⊢ ({1, 3} ∪ {1, 8}) = ({1, 3} ∪ ({1} ∪ {8}))
9		df-tp 4582	. 2 ⊢ {1, 3, 8} = ({1, 3} ∪ {8})
10	6, 8, 9	3eqtr4i 2775	1 ⊢ ({1, 3} ∪ {1, 8}) = {1, 3, 8}

Syntax hints:   = wceq 1541   ∪ cun 3899   ⊆ wss 3901  {csn 4577  {cpr 4579  {ctp 4581  1c1 10977  3c3 12134  8c8 12139

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3444  df-un 3906  df-in 3908  df-ss 3918  df-pr 4580  df-tp 4582

This theorem is referenced by:  ex-uni  29077