Theorem rexn0 3653

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 3557	. . 3 ⊢ (x ∈ A → A ≠ ∅)
2	1	a1d 22	. 2 ⊢ (x ∈ A → (φ → A ≠ ∅))
3	2	rexlimiv 2733	1 ⊢ (∃x ∈ A φ → A ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 1710   ≠ wne 2517  ∃wrex 2616  ∅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-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is referenced by: (None)