Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  snsssn Structured version   GIF version

Theorem snsssn 3523
 Description: If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.)
Hypothesis
Ref Expression
sneqr.1 A V
Assertion
Ref Expression
snsssn ({A} ⊆ {B} → A = B)

Proof of Theorem snsssn
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 dfss2 2928 . . 3 ({A} ⊆ {B} ↔ x(x {A} → x {B}))
2 elsn 3382 . . . . 5 (x {A} ↔ x = A)
3 elsn 3382 . . . . 5 (x {B} ↔ x = B)
42, 3imbi12i 228 . . . 4 ((x {A} → x {B}) ↔ (x = Ax = B))
54albii 1356 . . 3 (x(x {A} → x {B}) ↔ x(x = Ax = B))
61, 5bitri 173 . 2 ({A} ⊆ {B} ↔ x(x = Ax = B))
7 sneqr.1 . . 3 A V
8 sbceqal 2808 . . 3 (A V → (x(x = Ax = B) → A = B))
97, 8ax-mp 7 . 2 (x(x = Ax = B) → A = B)
106, 9sylbi 114 1 ({A} ⊆ {B} → A = B)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1240   = wceq 1242   ∈ wcel 1390  Vcvv 2551   ⊆ 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-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-v 2553  df-sbc 2759  df-in 2918  df-ss 2925  df-sn 3373 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator