Theorem tpidm12 4761

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 4645	. . 3 ⊢ {𝐴} = {𝐴, 𝐴}
2	1	uneq1i 4175	. 2 ⊢ ({𝐴} ∪ {𝐵}) = ({𝐴, 𝐴} ∪ {𝐵})
3		df-pr 4635	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
4		df-tp 4637	. 2 ⊢ {𝐴, 𝐴, 𝐵} = ({𝐴, 𝐴} ∪ {𝐵})
5	2, 3, 4	3eqtr4ri 2775	1 ⊢ {𝐴, 𝐴, 𝐵} = {𝐴, 𝐵}

Syntax hints:   = wceq 1538   ∪ cun 3962  {csn 4632  {cpr 4634  {ctp 4636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1541  df-ex 1778  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-un 3969  df-pr 4635  df-tp 4637

This theorem is referenced by:  tpidm13  4762  tpidm23  4763  tpidm  4764  fntpb  7233  hashtpg  14527  hash3tpde  14535