Theorem tpidm12 4727

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 4610	. . 3 ⊢ {𝐴} = {𝐴, 𝐴}
2	1	uneq1i 4135	. 2 ⊢ ({𝐴} ∪ {𝐵}) = ({𝐴, 𝐴} ∪ {𝐵})
3		df-pr 4600	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
4		df-tp 4602	. 2 ⊢ {𝐴, 𝐴, 𝐵} = ({𝐴, 𝐴} ∪ {𝐵})
5	2, 3, 4	3eqtr4ri 2764	1 ⊢ {𝐴, 𝐴, 𝐵} = {𝐴, 𝐵}

Syntax hints:   = wceq 1540   ∪ cun 3920  {csn 4597  {cpr 4599  {ctp 4601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3457  df-un 3927  df-pr 4600  df-tp 4602

This theorem is referenced by:  tpidm13  4728  tpidm23  4729  tpidm  4730  fntpb  7190  hashtpg  14460  hash3tpde  14468