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Theorem ssn0 3375
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 3374 . . . 4 ((𝐴𝐵𝐵 = ∅) → 𝐴 = ∅)
21ex 114 . . 3 (𝐴𝐵 → (𝐵 = ∅ → 𝐴 = ∅))
32necon3d 2329 . 2 (𝐴𝐵 → (𝐴 ≠ ∅ → 𝐵 ≠ ∅))
43imp 123 1 ((𝐴𝐵𝐴 ≠ ∅) → 𝐵 ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wne 2285  wss 3041  c0 3333
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-v 2662  df-dif 3043  df-in 3047  df-ss 3054  df-nul 3334
This theorem is referenced by: (None)
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