Theorem snsssn 2474

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} (_ {B} -> A = B)

Proof of Theorem snsssn

Step	Hyp	Ref	Expression
1		sssn 2469	. 2 |- ({A} (_ {B} <-> ({A} = (/) \/ {A} = {B}))
2		sneqr.1	. . . . . 6 |- A e. V
3	2	snnz 2454	. . . . 5 |- {A} =/= (/)
4		df-ne 1584	. . . . 5 |- ({A} =/= (/) <-> -. {A} = (/))
5	3, 4	mpbi 189	. . . 4 |- -. {A} = (/)
6	5	pm2.21i 77	. . 3 |- ({A} = (/) -> A = B)
7	2	sneqr 2473	. . 3 |- ({A} = {B} -> A = B)
8	6, 7	jaoi 341	. 2 |- (({A} = (/) \/ {A} = {B}) -> A = B)
9	1, 8	sylbi 199	1 |- ({A} (_ {B} -> A = B)

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   = wceq 954   e. wcel 956   =/= wne 1582  Vcvv 1807   (_ wss 2043  (/)c0 2276  {csn 2405

This theorem is referenced by:  pjspansnt 9440

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-sn 2408  df-pr 2409

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