Theorem rext 2754

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 1813	. . . 4 |- x e. V
2	1	snid 2435	. . 3 |- x e. {x}
3		snex 2750	. . . 4 |- {x} e. V
4		eleq2 1535	. . . . 5 |- (z = {x} -> (x e. z <-> x e. {x}))
5		eleq2 1535	. . . . 5 |- (z = {x} -> (y e. z <-> y e. {x}))
6	4, 5	imbi12d 626	. . . 4 |- (z = {x} -> ((x e. z -> y e. z) <-> (x e. {x} -> y e. {x})))
7	3, 6	cla4v 1868	. . 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 2421	. . 3 |- (y e. {x} <-> y = x)
10		equcomi 1128	. . 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 954   = wceq 956   e. wcel 958  {csn 2409

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-sep 2703  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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