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Theorem ss0b 3233
Description: Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23 and its converse. (Contributed by NM, 17-Sep-2003.)
Assertion
Ref Expression
ss0b (A ⊆ ∅ ↔ A = ∅)

Proof of Theorem ss0b
StepHypRef Expression
1 0ss 3232 . . 3 ∅ ⊆ A
2 eqss 2937 . . 3 (A = ∅ ↔ (A ⊆ ∅ ∅ ⊆ A))
31, 2mpbiran2 836 . 2 (A = ∅ ↔ A ⊆ ∅)
43bicomi 123 1 (A ⊆ ∅ ↔ A = ∅)
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1228  wss 2894  c0 3201
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-dif 2897  df-in 2901  df-ss 2908  df-nul 3202
This theorem is referenced by:  ss0  3234  un00  3240  ssdisj  3254  pw0  3485
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