Theorem eqsnOLD 4362

Description: Obsolete proof of eqsn 4361 as of 23-Jul-2021. (Contributed by NM, 15-Dec-2007.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
eqsnOLD	|- ( A =/= (/) -> ( A = { B } <-> A. x e. A x = B ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem eqsnOLD

Step	Hyp	Ref	Expression
1		eqimss 3657	. . 3 |- ( A = { B } -> A C_ { B } )
2		df-ne 2795	. . . . 5 |- ( A =/= (/) <-> -. A = (/) )
3		sssn 4358	. . . . . . 7 |- ( A C_ { B } <-> ( A = (/) \/ A = { B } ) )
4	3	biimpi 206	. . . . . 6 |- ( A C_ { B } -> ( A = (/) \/ A = { B } ) )
5	4	ord 392	. . . . 5 |- ( A C_ { B } -> ( -. A = (/) -> A = { B } ) )
6	2, 5	syl5bi 232	. . . 4 |- ( A C_ { B } -> ( A =/= (/) -> A = { B } ) )
7	6	com12 32	. . 3 |- ( A =/= (/) -> ( A C_ { B } -> A = { B } ) )
8	1, 7	impbid2 216	. 2 |- ( A =/= (/) -> ( A = { B } <-> A C_ { B } ) )
9		dfss3 3592	. . 3 |- ( A C_ { B } <-> A. x e. A x e. { B } )
10		velsn 4193	. . . 4 |- ( x e. { B } <-> x = B )
11	10	ralbii 2980	. . 3 |- ( A. x e. A x e. { B } <-> A. x e. A x = B )
12	9, 11	bitri 264	. 2 |- ( A C_ { B } <-> A. x e. A x = B )
13	8, 12	syl6bb 276	1 |- ( A =/= (/) -> ( A = { B } <-> A. x e. A x = B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   C_ wss 3574   (/)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-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178

This theorem is referenced by: (None)