Theorem axpr 2834

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 2835 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 2833	. . 3 |- {x, y} e. V
2	1	isseti 1862	. 2 |- E.z z = {x, y}
3		dfcleq 1516	. . . 4 |- (z = {x, y} <-> A.w(w e. z <-> w e. {x, y}))
4		visset 1860	. . . . . . . 8 |- w e. V
5	4	elpr 2476	. . . . . . 7 |- (w e. {x, y} <-> (w = x \/ w = y))
6	5	bibi2i 619	. . . . . 6 |- ((w e. z <-> w e. {x, y}) <-> (w e. z <-> (w = x \/ w = y)))
7		bi2 156	. . . . . 6 |- ((w e. z <-> (w = x \/ w = y)) -> ((w = x \/ w = y) -> w e. z))
8	6, 7	sylbi 206	. . . . 5 |- ((w e. z <-> w e. {x, y}) -> ((w = x \/ w = y) -> w e. z))
9	8	19.20i 1033	. . . 4 |- (A.w(w e. z <-> w e. {x, y}) -> A.w((w = x \/ w = y) -> w e. z))
10	3, 9	sylbi 206	. . 3 |- (z = {x, y} -> A.w((w = x \/ w = y) -> w e. z))
11	10	19.22i 1081	. 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 153   \/ wo 229  A.wal 995   = wceq 997   e. wcel 999  E.wex 1021  {cpr 2462

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-rep 2748  ax-sep 2758  ax-nul 2765  ax-pow 2798

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-v 1859  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-pw 2454  df-sn 2464  df-pr 2465

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