Theorem sssn 2471

Description: The subsets of a singleton.

Assertion

Ref	Expression
sssn	|- (A (_ {B} <-> (A = (/) \/ A = {B}))

Proof of Theorem sssn

Step	Hyp	Ref	Expression
1		ssel 2061	. . . . . . . . . . 11 |- (A (_ {B} -> (x e. A -> x e. {B}))
2		elsni 2430	. . . . . . . . . . 11 |- (x e. {B} -> x = B)
3	1, 2	syl6 22	. . . . . . . . . 10 |- (A (_ {B} -> (x e. A -> x = B))
4		eleq1 1533	. . . . . . . . . 10 |- (x = B -> (x e. A <-> B e. A))
5	3, 4	syl6 22	. . . . . . . . 9 |- (A (_ {B} -> (x e. A -> (x e. A <-> B e. A)))
6	5	ibd 593	. . . . . . . 8 |- (A (_ {B} -> (x e. A -> B e. A))
7	6	19.23adv 1214	. . . . . . 7 |- (A (_ {B} -> (E.x x e. A -> B e. A))
8		n0 2287	. . . . . . 7 |- (-. A = (/) <-> E.x x e. A)
9	7, 8	syl5ib 206	. . . . . 6 |- (A (_ {B} -> (-. A = (/) -> B e. A))
10		snssi 2464	. . . . . 6 |- (B e. A -> {B} (_ A)
11	9, 10	syl6 22	. . . . 5 |- (A (_ {B} -> (-. A = (/) -> {B} (_ A))
12	11	anc2li 302	. . . 4 |- (A (_ {B} -> (-. A = (/) -> (A (_ {B} /\ {B} (_ A)))
13		eqss 2075	. . . 4 |- (A = {B} <-> (A (_ {B} /\ {B} (_ A))
14	12, 13	syl6ibr 213	. . 3 |- (A (_ {B} -> (-. A = (/) -> A = {B}))
15	14	orrd 233	. 2 |- (A (_ {B} -> (A = (/) \/ A = {B}))
16		0ss 2299	. . . 4 |- (/) (_ {B}
17		sseq1 2080	. . . 4 |- (A = (/) -> (A (_ {B} <-> (/) (_ {B}))
18	16, 17	mpbiri 194	. . 3 |- (A = (/) -> A (_ {B})
19		eqimss 2107	. . 3 |- (A = {B} -> A (_ {B})
20	18, 19	jaoi 341	. 2 |- ((A = (/) \/ A = {B}) -> A (_ {B})
21	15, 20	impbi 157	1 |- (A (_ {B} <-> (A = (/) \/ A = {B}))

Syntax hints:  -. wn 2   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 955   e. wcel 957  E.wex 979   (_ wss 2045  (/)c0 2278  {csn 2407

This theorem is referenced by:  eqsn 2472  sspr 2473  snsssn 2476  pwsn 2498  foconst 3680  0top 7614  sn0top 7626

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-sn 2410  df-pr 2411

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