Theorem axpr 2778

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 2779 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 2777	. . 3 |- {x, y} e. V
2	1	isseti 1815	. 2 |- E.z z = {x, y}
3		dfcleq 1470	. . . 4 |- (z = {x, y} <-> A.w(w e. z <-> w e. {x, y}))
4		visset 1813	. . . . . . . 8 |- w e. V
5	4	elpr 2424	. . . . . . 7 |- (w e. {x, y} <-> (w = x \/ w = y))
6	5	bibi2i 608	. . . . . 6 |- ((w e. z <-> w e. {x, y}) <-> (w e. z <-> (w = x \/ w = y)))
7		bi2 149	. . . . . 6 |- ((w e. z <-> (w = x \/ w = y)) -> ((w = x \/ w = y) -> w e. z))
8	6, 7	sylbi 199	. . . . 5 |- ((w e. z <-> w e. {x, y}) -> ((w = x \/ w = y) -> w e. z))
9	8	19.20i 992	. . . 4 |- (A.w(w e. z <-> w e. {x, y}) -> A.w((w = x \/ w = y) -> w e. z))
10	3, 9	sylbi 199	. . 3 |- (z = {x, y} -> A.w((w = x \/ w = y) -> w e. z))
11	10	19.22i 1040	. 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 146   \/ wo 222  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  {cpr 2410

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-11 967  ax-12 968  ax-13 969  ax-14 970  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  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413

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