Theorem tpidm12 3821

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 3747	. . 3 |- {A} = {A, A}
2	1	uneq1i 3414	. 2 |- ({A} u. {B}) = ({A, A} u. {B})
3		df-pr 3742	. 2 |- {A, B} = ({A} u. {B})
4		df-tp 3743	. 2 |- {A, A, B} = ({A, A} u. {B})
5	2, 3, 4	3eqtr4ri 2384	1 |- {A, A, B} = {A, B}

Syntax hints:   = wceq 1642   u. cun 3207  {csn 3737  {cpr 3738  {ctp 3739

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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-pr 3742  df-tp 3743

This theorem is referenced by:  tpidm13  3822  tpidm23  3823  tpidm  3824