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Theorem sseq0 3286
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 2995 . . 3 (𝐵 = ∅ → (𝐴𝐵𝐴 ⊆ ∅))
2 ss0 3285 . . 3 (𝐴 ⊆ ∅ → 𝐴 = ∅)
31, 2syl6bi 156 . 2 (𝐵 = ∅ → (𝐴𝐵𝐴 = ∅))
43impcom 120 1 ((𝐴𝐵𝐵 = ∅) → 𝐴 = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wss 2945  c0 3252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-dif 2948  df-in 2952  df-ss 2959  df-nul 3253
This theorem is referenced by:  ssn0  3287  ssdifin0  3332
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