Theorem ss0b 3581

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 3580	. . 3 ⊢ ∅ ⊆ A
2		eqss 3288	. . 3 ⊢ (A = ∅ ↔ (A ⊆ ∅ ∧ ∅ ⊆ A))
3	1, 2	mpbiran2 885	. 2 ⊢ (A = ∅ ↔ A ⊆ ∅)
4	3	bicomi 193	1 ⊢ (A ⊆ ∅ ↔ A = ∅)

Syntax hints:   ↔ wb 176   = wceq 1642   ⊆ wss 3258  ∅c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552

This theorem is referenced by:  ss0  3582  un00  3587  ssdisj  3601  pw0  4161  ssfin  4471