Theorem rexn0 3652

Description: Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)

Assertion

Ref	Expression
rexn0	⊢ (∃x ∈ A φ → A ≠ ∅)

Distinct variable group:   x,A

Allowed substitution hint:   φ(x)

Proof of Theorem rexn0

Step	Hyp	Ref	Expression
1		ne0i 3556	. . 3 ⊢ (x ∈ A → A ≠ ∅)
2	1	a1d 22	. 2 ⊢ (x ∈ A → (φ → A ≠ ∅))
3	2	rexlimiv 2732	1 ⊢ (∃x ∈ A φ → A ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 1710   ≠ wne 2516  ∃wrex 2615  ∅c0 3550

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551

This theorem is referenced by: (None)