Theorem dfpr2 2418

Description: Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15.

Assertion

Ref	Expression
dfpr2	|- {A, B} = {x | (x = A \/ x = B)}

Distinct variable groups:   x,A   x,B

Proof of Theorem dfpr2

Step	Hyp	Ref	Expression
1		df-pr 2409	. 2 |- {A, B} = ({A} u. {B})
2		elun 2169	. . . 4 |- (x e. ({A} u. {B}) <-> (x e. {A} \/ x e. {B}))
3		elsn 2417	. . . . 5 |- (x e. {A} <-> x = A)
4		elsn 2417	. . . . 5 |- (x e. {B} <-> x = B)
5	3, 4	orbi12i 257	. . . 4 |- ((x e. {A} \/ x e. {B}) <-> (x = A \/ x = B))
6	2, 5	bitr 173	. . 3 |- (x e. ({A} u. {B}) <-> (x = A \/ x = B))
7	6	abbi2i 1571	. 2 |- ({A} u. {B}) = {x | (x = A \/ x = B)}
8	1, 7	eqtr 1492	1 |- {A, B} = {x | (x = A \/ x = B)}

Syntax hints:   \/ wo 222   = wceq 954   e. wcel 956  {cab 1461   u. cun 2041  {csn 2405  {cpr 2406

This theorem is referenced by:  elprg 2419  pwpw0 2465  pwsn 2496  zfpair 2772  grothprimlem 8721

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-un 2046  df-sn 2408  df-pr 2409

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