Theorem snsssn 3148

Description: If a singleton is a subset of another, their members are equal.

Hypothesis

Ref	Expression
sneqr.1	|- A e. _V

Assertion

Ref	Expression
snsssn	|- ({A} C_ {B} -> A = B)

Proof of Theorem snsssn

Step	Hyp	Ref	Expression
1		sssn 3142	. 2 |- ({A} C_ {B} <-> ({A} = (/) \/ {A} = {B}))
2		sneqr.1	. . . . . 6 |- A e. _V
3	2	snnz 3119	. . . . 5 |- {A} =/= (/)
4		df-ne 2019	. . . . 5 |- ({A} =/= (/) <-> -. {A} = (/))
5	3, 4	mpbi 206	. . . 4 |- -. {A} = (/)
6	5	pm2.21i 93	. . 3 |- ({A} = (/) -> A = B)
7	2	sneqr 3147	. . 3 |- ({A} = {B} -> A = B)
8	6, 7	jaoi 368	. 2 |- (({A} = (/) \/ {A} = {B}) -> A = B)
9	1, 8	sylbi 216	1 |- ({A} C_ {B} -> A = B)

Syntax hints:  -. wn 2   -> wi 3   \/ wo 239   = wceq 1298   e. wcel 1300   =/= wne 2017  _Vcvv 2292   C_ wss 2593  (/)c0 2875  {csn 3044

This theorem is referenced by:  pjspansn 11133

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-in 2603  df-ss 2605  df-nul 2876  df-sn 3049

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