Theorem ssn0 3584

Description: A class with a nonempty subclass is nonempty. (Contributed by NM, 17-Feb-2007.)

Assertion

Ref	Expression
ssn0	⊢ ((A ⊆ B ∧ A ≠ ∅) → B ≠ ∅)

Proof of Theorem ssn0

Step	Hyp	Ref	Expression
1		sseq0 3583	. . . 4 ⊢ ((A ⊆ B ∧ B = ∅) → A = ∅)
2	1	ex 423	. . 3 ⊢ (A ⊆ B → (B = ∅ → A = ∅))
3	2	necon3d 2555	. 2 ⊢ (A ⊆ B → (A ≠ ∅ → B ≠ ∅))
4	3	imp 418	1 ⊢ ((A ⊆ B ∧ A ≠ ∅) → B ≠ ∅)

This proof depends on syntax axioms:   → wi 4   ∧ wa 358   = wceq 1642   ≠ wne 2517   ⊆ wss 3258  ∅c0 3551

This proof depends on 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 proof 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-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552

This theorem is used by: (None)