Theorem sssnm 3499

Description: The inhabited subset of a singleton. (Contributed by Jim Kingdon, 10-Aug-2018.)

Assertion

Ref	Expression
sssnm	⊢ (∃x x ∈ A → (A ⊆ {B} ↔ A = {B}))

Distinct variable groups:   x,A   x,B

Proof of Theorem sssnm

Step	Hyp	Ref	Expression
1		ssel 2916	. . . . . . . . . 10 ⊢ (A ⊆ {B} → (x ∈ A → x ∈ {B}))
2		elsni 3374	. . . . . . . . . 10 ⊢ (x ∈ {B} → x = B)
3	1, 2	syl6 29	. . . . . . . . 9 ⊢ (A ⊆ {B} → (x ∈ A → x = B))
4		eleq1 2082	. . . . . . . . 9 ⊢ (x = B → (x ∈ A ↔ B ∈ A))
5	3, 4	syl6 29	. . . . . . . 8 ⊢ (A ⊆ {B} → (x ∈ A → (x ∈ A ↔ B ∈ A)))
6	5	ibd 167	. . . . . . 7 ⊢ (A ⊆ {B} → (x ∈ A → B ∈ A))
7	6	exlimdv 1682	. . . . . 6 ⊢ (A ⊆ {B} → (∃x x ∈ A → B ∈ A))
8		snssi 3482	. . . . . 6 ⊢ (B ∈ A → {B} ⊆ A)
9	7, 8	syl6 29	. . . . 5 ⊢ (A ⊆ {B} → (∃x x ∈ A → {B} ⊆ A))
10	9	anc2li 312	. . . 4 ⊢ (A ⊆ {B} → (∃x x ∈ A → (A ⊆ {B} ∧ {B} ⊆ A)))
11		eqss 2937	. . . 4 ⊢ (A = {B} ↔ (A ⊆ {B} ∧ {B} ⊆ A))
12	10, 11	syl6ibr 151	. . 3 ⊢ (A ⊆ {B} → (∃x x ∈ A → A = {B}))
13	12	com12 27	. 2 ⊢ (∃x x ∈ A → (A ⊆ {B} → A = {B}))
14		eqimss 2974	. 2 ⊢ (A = {B} → A ⊆ {B})
15	13, 14	impbid1 130	1 ⊢ (∃x x ∈ A → (A ⊆ {B} ↔ A = {B}))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1228  ∃wex 1362   ∈ wcel 1374   ⊆ wss 2894  {csn 3350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-in 2901  df-ss 2908  df-sn 3356

This theorem is referenced by:  eqsnm  3500