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Theorem difprsnss 3493
 Description: Removal of a singleton from an unordered pair. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
difprsnss ({A, B} ∖ {A}) ⊆ {B}

Proof of Theorem difprsnss
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 vex 2554 . . . . 5 x V
21elpr 3385 . . . 4 (x {A, B} ↔ (x = A x = B))
3 elsn 3382 . . . . 5 (x {A} ↔ x = A)
43notbii 593 . . . 4 x {A} ↔ ¬ x = A)
5 biorf 662 . . . . 5 x = A → (x = B ↔ (x = A x = B)))
65biimparc 283 . . . 4 (((x = A x = B) ¬ x = A) → x = B)
72, 4, 6syl2anb 275 . . 3 ((x {A, B} ¬ x {A}) → x = B)
8 eldif 2921 . . 3 (x ({A, B} ∖ {A}) ↔ (x {A, B} ¬ x {A}))
9 elsn 3382 . . 3 (x {B} ↔ x = B)
107, 8, 93imtr4i 190 . 2 (x ({A, B} ∖ {A}) → x {B})
1110ssriv 2943 1 ({A, B} ∖ {A}) ⊆ {B}
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 97   ∨ wo 628   = wceq 1242   ∈ wcel 1390   ∖ cdif 2908   ⊆ wss 2911  {csn 3367  {cpr 3368 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-in1 544  ax-in2 545  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-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374 This theorem is referenced by: (None)
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