Theorem tpidm12 4714

Description: Unordered triple {𝐴, 𝐴, 𝐵} is just an overlong way to write {𝐴, 𝐵}. (Contributed by David A. Wheeler, 10-May-2015.)

Assertion

Ref	Expression
tpidm12	⊢ {𝐴, 𝐴, 𝐵} = {𝐴, 𝐵}

Proof of Theorem tpidm12

Step	Hyp	Ref	Expression
1		dfsn2 4595	. . 3 ⊢ {𝐴} = {𝐴, 𝐴}
2	1	uneq1i 4117	. 2 ⊢ ({𝐴} ∪ {𝐵}) = ({𝐴, 𝐴} ∪ {𝐵})
3		df-pr 4585	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
4		df-tp 4587	. 2 ⊢ {𝐴, 𝐴, 𝐵} = ({𝐴, 𝐴} ∪ {𝐵})
5	2, 3, 4	3eqtr4ri 2796	1 ⊢ {𝐴, 𝐴, 𝐵} = {𝐴, 𝐵}

Syntax hints:   = wceq 1560   ∪ cun 3902  {csn 4582  {cpr 4584  {ctp 4586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-un 3909  df-pr 4585  df-tp 4587

This theorem is referenced by:  tpidm13  4715  tpidm23  4716  tpidm  4717  fntpb  7193  hashtpg  14498  hash3tpde  14506