Theorem ss0b 3534

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

Step	Hyp	Ref	Expression
1		0ss 3533	. . 3 ⊢ ∅ ⊆ 𝐴
2		eqss 3242	. . 3 ⊢ (𝐴 = ∅ ↔ (𝐴 ⊆ ∅ ∧ ∅ ⊆ 𝐴))
3	1, 2	mpbiran2 949	. 2 ⊢ (𝐴 = ∅ ↔ 𝐴 ⊆ ∅)
4	3	bicomi 132	1 ⊢ (𝐴 ⊆ ∅ ↔ 𝐴 = ∅)

Syntax hints:   ↔ wb 105   = wceq 1397   ⊆ wss 3200  ∅c0 3494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213  df-nul 3495

This theorem is referenced by:  ss0  3535  un00  3541  ssdisj  3551  pw0  3820  card0  7391  0nnei  14876