Theorem rext 2744

Description: A theorem similar to extensionality, requiring the existence of a singleton. Exercise 8 of [TakeutiZaring] p. 16.

Assertion

Ref	Expression
rext	|- (A.z(x e. z -> y e. z) -> x = y)

Distinct variable group:   x,y,z

Proof of Theorem rext

Step	Hyp	Ref	Expression
1		visset 1804	. . . 4 |- x e. V
2	1	snid 2425	. . 3 |- x e. {x}
3		snex 2740	. . . 4 |- {x} e. V
4		eleq2 1527	. . . . 5 |- (z = {x} -> (x e. z <-> x e. {x}))
5		eleq2 1527	. . . . 5 |- (z = {x} -> (y e. z <-> y e. {x}))
6	4, 5	imbi12d 624	. . . 4 |- (z = {x} -> ((x e. z -> y e. z) <-> (x e. {x} -> y e. {x})))
7	3, 6	cla4v 1859	. . 3 |- (A.z(x e. z -> y e. z) -> (x e. {x} -> y e. {x}))
8	2, 7	mpi 44	. 2 |- (A.z(x e. z -> y e. z) -> y e. {x})
9		elsn 2411	. . 3 |- (y e. {x} <-> y = x)
10		equcomi 1124	. . 3 |- (y = x -> x = y)
11	9, 10	sylbi 199	. 2 |- (y e. {x} -> x = y)
12	8, 11	syl 10	1 |- (A.z(x e. z -> y e. z) -> x = y)

Syntax hints:   -> wi 3  A.wal 951   = wceq 953   e. wcel 955  {csn 2399

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403

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