Theorem sssnm 3516

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 2933	. . . . . . . . . 10 ⊢ (A ⊆ {B} → (x ∈ A → x ∈ {B}))
2		elsni 3391	. . . . . . . . . 10 ⊢ (x ∈ {B} → x = B)
3	1, 2	syl6 29	. . . . . . . . 9 ⊢ (A ⊆ {B} → (x ∈ A → x = B))
4		eleq1 2097	. . . . . . . . 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 1697	. . . . . 6 ⊢ (A ⊆ {B} → (∃x x ∈ A → B ∈ A))
8		snssi 3499	. . . . . 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 2954	. . . 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 2991	. 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 1242  ∃wex 1378   ∈ wcel 1390   ⊆ wss 2911  {csn 3367

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-sn 3373

This theorem is referenced by:  eqsnm  3517