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Theorem ssn0 3457
Description: A class with a nonempty subclass is nonempty. (Contributed by NM, 17-Feb-2007.)
Assertion
Ref Expression
ssn0 ((𝐴𝐵𝐴 ≠ ∅) → 𝐵 ≠ ∅)

Proof of Theorem ssn0
StepHypRef Expression
1 sseq0 3456 . . . 4 ((𝐴𝐵𝐵 = ∅) → 𝐴 = ∅)
21ex 114 . . 3 (𝐴𝐵 → (𝐵 = ∅ → 𝐴 = ∅))
32necon3d 2384 . 2 (𝐴𝐵 → (𝐴 ≠ ∅ → 𝐵 ≠ ∅))
43imp 123 1 ((𝐴𝐵𝐴 ≠ ∅) → 𝐵 ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1348  wne 2340  wss 3121  c0 3414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415
This theorem is referenced by: (None)
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