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Theorem sseq0 3252
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 ((AB B = ∅) → A = ∅)

Proof of Theorem sseq0
StepHypRef Expression
1 sseq2 2961 . . 3 (B = ∅ → (ABA ⊆ ∅))
2 ss0 3251 . . 3 (A ⊆ ∅ → A = ∅)
31, 2syl6bi 152 . 2 (B = ∅ → (ABA = ∅))
43impcom 116 1 ((AB B = ∅) → A = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  wss 2911  c0 3218
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-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-dif 2914  df-in 2918  df-ss 2925  df-nul 3219
This theorem is referenced by:  ssn0  3253  ssdifin0  3298
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