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Theorem sseq0 3464
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 3179 . . 3 (𝐵 = ∅ → (𝐴𝐵𝐴 ⊆ ∅))
2 ss0 3463 . . 3 (𝐴 ⊆ ∅ → 𝐴 = ∅)
31, 2syl6bi 163 . 2 (𝐵 = ∅ → (𝐴𝐵𝐴 = ∅))
43impcom 125 1 ((𝐴𝐵𝐵 = ∅) → 𝐴 = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wss 3129  c0 3422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-dif 3131  df-in 3135  df-ss 3142  df-nul 3423
This theorem is referenced by:  ssn0  3465  ssdifin0  3504  fieq0  6972  fisumss  11393  strleund  12554  strleun  12555
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