Theorem tpidm12 4712

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 4593	. . 3 ⊢ {𝐴} = {𝐴, 𝐴}
2	1	uneq1i 4162	. 2 ⊢ ({𝐴} ∪ {𝐵}) = ({𝐴, 𝐴} ∪ {𝐵})
3		df-pr 4587	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
4		df-tp 4589	. 2 ⊢ {𝐴, 𝐴, 𝐵} = ({𝐴, 𝐴} ∪ {𝐵})
5	2, 3, 4	3eqtr4ri 2794	1 ⊢ {𝐴, 𝐴, 𝐵} = {𝐴, 𝐵}

This proof depends on syntax axioms:   = wceq 1569   ∪ cun 4148  {csn 4584  {cpr 4586  {ctp 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-un 4150  df-pr 4587  df-tp 4589

This theorem is used by:  tpidm13  4713  tpidm23  4714  tpidm  4715  fntpb  7219  hashtpg  14637  hash3tpde  14645