Theorem ex-un 31055

Description: Example for df-un 4154. 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 4173	. . 3 ⊢ (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ ({1} ∪ {8}))
2		snsspr1 4773	. . . . 5 ⊢ {1} ⊆ {1, 3}
3		ssequn2 4190	. . . . 5 ⊢ ({1} ⊆ {1, 3} ↔ ({1, 3} ∪ {1}) = {1, 3})
4	2, 3	mpbi 233	. . . 4 ⊢ ({1, 3} ∪ {1}) = {1, 3}
5	4	uneq1i 4166	. . 3 ⊢ (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ {8})
6	1, 5	eqtr3i 2788	. 2 ⊢ ({1, 3} ∪ ({1} ∪ {8})) = ({1, 3} ∪ {8})
7		df-pr 4591	. . 3 ⊢ {1, 8} = ({1} ∪ {8})
8	7	uneq2i 4167	. 2 ⊢ ({1, 3} ∪ {1, 8}) = ({1, 3} ∪ ({1} ∪ {8}))
9		df-tp 4593	. 2 ⊢ {1, 3, 8} = ({1, 3} ∪ {8})
10	6, 8, 9	3eqtr4i 2796	1 ⊢ ({1, 3} ∪ {1, 8}) = {1, 3, 8}

This proof depends on syntax axioms:   = wceq 1570   ⊆ wss 3900   ∪ cun 4152  {csn 4588  {cpr 4590  {ctp 4592  1c1 11228  3c3 12425  8c8 12430

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-un 4154  df-pr 4591  df-tp 4593

This theorem is used by:  ex-uni  31057