Theorem tpidm12 4694

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 4577	. . 3 ⊢ {𝐴} = {𝐴, 𝐴}
2	1	uneq1i 4096	. 2 ⊢ ({𝐴} ∪ {𝐵}) = ({𝐴, 𝐴} ∪ {𝐵})
3		df-pr 4567	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
4		df-tp 4569	. 2 ⊢ {𝐴, 𝐴, 𝐵} = ({𝐴, 𝐴} ∪ {𝐵})
5	2, 3, 4	3eqtr4ri 2772	1 ⊢ {𝐴, 𝐴, 𝐵} = {𝐴, 𝐵}

Syntax hints:   = wceq 1537   ∪ cun 3887  {csn 4564  {cpr 4566  {ctp 4568

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 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1540  df-ex 1778  df-sb 2063  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3436  df-un 3894  df-pr 4567  df-tp 4569

This theorem is referenced by:  tpidm13  4695  tpidm23  4696  tpidm  4697  fntpb  7105  hashtpg  14227