Theorem tpidm12 4716

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 4597	. . 3 ⊢ {𝐴} = {𝐴, 𝐴}
2	1	uneq1i 4111	. 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 1570   ∪ cun 3897  {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 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-un 3904  df-pr 4587  df-tp 4589

This theorem is used by:  tpidm13  4717  tpidm23  4718  tpidm  4719  fntpb  7208  hashtpg  14550  hash3tpde  14558