Theorem prid1g 3104

Description: An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)

Assertion

Ref	Expression
prid1g	|- (A e. C -> A e. {A, B})

Proof of Theorem prid1g

Step	Hyp	Ref	Expression
1		eqid 1884	. . 3 |- A = A
2	1	orci 292	. 2 |- (A = A \/ A = B)
3		elprg 3060	. 2 |- (A e. C -> (A e. {A, B} <-> (A = A \/ A = B)))
4	2, 3	mpbiri 211	1 |- (A e. C -> A e. {A, B})

Syntax hints:   -> wi 3   \/ wo 239   = wceq 1298   e. wcel 1300  {cpr 3045

This theorem is referenced by:  prid2g 3105  prid1 3106  gcdcllem2 13719  gcdcllem3 13720

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-un 2600  df-sn 3049  df-pr 3050

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