Theorem tpidm12 3822

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 3748	. . 3 ⊢ {A} = {A, A}
2	1	uneq1i 3415	. 2 ⊢ ({A} ∪ {B}) = ({A, A} ∪ {B})
3		df-pr 3743	. 2 ⊢ {A, B} = ({A} ∪ {B})
4		df-tp 3744	. 2 ⊢ {A, A, B} = ({A, A} ∪ {B})
5	2, 3, 4	3eqtr4ri 2384	1 ⊢ {A, A, B} = {A, B}

Syntax hints:   = wceq 1642   ∪ cun 3208  {csn 3738  {cpr 3739  {ctp 3740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-pr 3743  df-tp 3744

This theorem is referenced by:  tpidm13  3823  tpidm23  3824  tpidm  3825