Theorem axpr 2746

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 2747 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 2745	. . 3 |- {x, y} e. V
2	1	isseti 1790	. 2 |- E.z z = {x, y}
3		dfcleq 1447	. . . 4 |- (z = {x, y} <-> A.w(w e. z <-> w e. {x, y}))
4		visset 1788	. . . . . . . 8 |- w e. V
5	4	elpr 2395	. . . . . . 7 |- (w e. {x, y} <-> (w = x \/ w = y))
6	5	bibi2i 606	. . . . . 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 968	. . . 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 1016	. 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 950  E.wex 956   = wceq 1099   e. wcel 1105  {cpr 2381

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-rep 2661  ax-sep 2671  ax-nul 2678  ax-pow 2710

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-v 1787  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-pw 2373  df-sn 2383  df-pr 2384

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