Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  sseq0 GIF version

Theorem sseq0 3258
 Description: A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
sseq0 ((𝐴𝐵𝐵 = ∅) → 𝐴 = ∅)

Proof of Theorem sseq0
StepHypRef Expression
1 sseq2 2967 . . 3 (𝐵 = ∅ → (𝐴𝐵𝐴 ⊆ ∅))
2 ss0 3257 . . 3 (𝐴 ⊆ ∅ → 𝐴 = ∅)
31, 2syl6bi 152 . 2 (𝐵 = ∅ → (𝐴𝐵𝐴 = ∅))
43impcom 116 1 ((𝐴𝐵𝐵 = ∅) → 𝐴 = ∅)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ⊆ wss 2917  ∅c0 3224 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-nul 3225 This theorem is referenced by:  ssn0  3259  ssdifin0  3304
 Copyright terms: Public domain W3C validator