Theorem dfpr2 2422

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 2413	. 2 |- {A, B} = ({A} u. {B})
2		elun 2173	. . . 4 |- (x e. ({A} u. {B}) <-> (x e. {A} \/ x e. {B}))
3		elsn 2421	. . . . 5 |- (x e. {A} <-> x = A)
4		elsn 2421	. . . . 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 1574	. 2 |- ({A} u. {B}) = {x | (x = A \/ x = B)}
8	1, 7	eqtr 1495	1 |- {A, B} = {x | (x = A \/ x = B)}

Syntax hints:   \/ wo 222   = wceq 956   e. wcel 958  {cab 1463   u. cun 2045  {csn 2409  {cpr 2410

This theorem is referenced by:  elprg 2423  pwpw0 2469  pwsn 2500  pwsnALT 2501  zfpair 2777  grothprimlem 8782

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413

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