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Theorem ss0b 3250
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 3249 . . 3 ∅ ⊆ A
2 eqss 2954 . . 3 (A = ∅ ↔ (A ⊆ ∅ ∅ ⊆ A))
31, 2mpbiran2 847 . 2 (A = ∅ ↔ A ⊆ ∅)
43bicomi 123 1 (A ⊆ ∅ ↔ A = ∅)
Colors of variables: wff set class
Syntax hints:  wb 98   = 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:  ss0  3251  un00  3257  ssdisj  3271  pw0  3502
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