Theorem sseq0 3583

Description: A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
sseq0	⊢ ((A ⊆ B ∧ B = ∅) → A = ∅)

Proof of Theorem sseq0

Step	Hyp	Ref	Expression
1		sseq2 3294	. . 3 ⊢ (B = ∅ → (A ⊆ B ↔ A ⊆ ∅))
2		ss0 3582	. . 3 ⊢ (A ⊆ ∅ → A = ∅)
3	1, 2	syl6bi 219	. 2 ⊢ (B = ∅ → (A ⊆ B → A = ∅))
4	3	impcom 419	1 ⊢ ((A ⊆ B ∧ B = ∅) → A = ∅)

Syntax hints:   → wi 4   ∧ wa 358   = 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:  ssn0  3584  ssdifin0  3632