Theorem axpr 3523

Description: Unabbreviated version of the Axiom of Pairing of ZF set theory, derived as a theorem from the other axioms.

This theorem should not be referenced by any proof. Instead, use ax-pr 3524 below so that the uses of the Axiom of Pairing can be more easily identified.

Assertion

Ref	Expression
axpr	|- E.zA.w((w = x \/ w = y) -> w e. z)

Distinct variable groups:   x,z,w   y,z,w

Proof of Theorem axpr

Step	Hyp	Ref	Expression
1		zfpair 3522	. . 3 |- {x, y} e. _V
2	1	isseti 2297	. 2 |- E.z z = {x, y}
3		dfcleq 1878	. . . 4 |- (z = {x, y} <-> A.w(w e. z <-> w e. {x, y}))
4		visset 2295	. . . . . . . 8 |- w e. _V
5	4	elpr 3061	. . . . . . 7 |- (w e. {x, y} <-> (w = x \/ w = y))
6	5	bibi2i 669	. . . . . 6 |- ((w e. z <-> w e. {x, y}) <-> (w e. z <-> (w = x \/ w = y)))
7		bi2 166	. . . . . 6 |- ((w e. z <-> (w = x \/ w = y)) -> ((w = x \/ w = y) -> w e. z))
8	6, 7	sylbi 216	. . . . 5 |- ((w e. z <-> w e. {x, y}) -> ((w = x \/ w = y) -> w e. z))
9	8	alimi 1338	. . . 4 |- (A.w(w e. z <-> w e. {x, y}) -> A.w((w = x \/ w = y) -> w e. z))
10	3, 9	sylbi 216	. . 3 |- (z = {x, y} -> A.w((w = x \/ w = y) -> w e. z))
11	10	eximi 1387	. 2 |- (E.z z = {x, y} -> E.zA.w((w = x \/ w = y) -> w e. z))
12	2, 11	ax-mp 7	1 |- E.zA.w((w = x \/ w = y) -> w e. z)

Syntax hints:   -> wi 3   <-> wb 163   \/ wo 239  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  {cpr 3045

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-14 1312  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  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481

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-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050

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