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Theorem eqsnm 3516
 Description: Two ways to express that an inhabited set equals a singleton. (Contributed by Jim Kingdon, 11-Aug-2018.)
Assertion
Ref Expression
eqsnm (x x A → (A = {B} ↔ x A x = B))
Distinct variable groups:   x,A   x,B

Proof of Theorem eqsnm
StepHypRef Expression
1 dfss3 2929 . . 3 (A ⊆ {B} ↔ x A x {B})
2 elsn 3381 . . . 4 (x {B} ↔ x = B)
32ralbii 2324 . . 3 (x A x {B} ↔ x A x = B)
41, 3bitri 173 . 2 (A ⊆ {B} ↔ x A x = B)
5 sssnm 3515 . 2 (x x A → (A ⊆ {B} ↔ A = {B}))
64, 5syl5rbbr 184 1 (x x A → (A = {B} ↔ x A x = B))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∀wral 2300   ⊆ wss 2911  {csn 3366 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-bnd 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-ral 2305  df-v 2553  df-in 2918  df-ss 2925  df-sn 3372 This theorem is referenced by: (None)
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