Theorem issn 4363

Description: A sufficient condition for a (nonempty) set to be a singleton. (Contributed by AV, 20-Sep-2020.)

Assertion

Ref	Expression
issn	|- ( E. x e. A A. y e. A x = y -> E. z A = { z } )

Distinct variable group:   x, A, y, z

Proof of Theorem issn

Step	Hyp	Ref	Expression
1		equcom 1945	. . . . . 6 |- ( x = y <-> y = x )
2	1	a1i 11	. . . . 5 |- ( x e. A -> ( x = y <-> y = x ) )
3	2	ralbidv 2986	. . . 4 |- ( x e. A -> ( A. y e. A x = y <-> A. y e. A y = x ) )
4		ne0i 3921	. . . . 5 |- ( x e. A -> A =/= (/) )
5		eqsn 4361	. . . . 5 |- ( A =/= (/) -> ( A = { x } <-> A. y e. A y = x ) )
6	4, 5	syl 17	. . . 4 |- ( x e. A -> ( A = { x } <-> A. y e. A y = x ) )
7	3, 6	bitr4d 271	. . 3 |- ( x e. A -> ( A. y e. A x = y <-> A = { x } ) )
8		sneq 4187	. . . . 5 |- ( z = x -> { z } = { x } )
9	8	eqeq2d 2632	. . . 4 |- ( z = x -> ( A = { z } <-> A = { x } ) )
10	9	spcegv 3294	. . 3 |- ( x e. A -> ( A = { x } -> E. z A = { z } ) )
11	7, 10	sylbid 230	. 2 |- ( x e. A -> ( A. y e. A x = y -> E. z A = { z } ) )
12	11	rexlimiv 3027	1 |- ( E. x e. A A. y e. A x = y -> E. z A = { z } )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   (/)c0 3915   {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178

This theorem is referenced by:  n0snor2el  4364