Theorem rexn0 3507

Description: Restricted existential quantification implies its restriction is nonempty (it is also inhabited as shown in rexm 3508). (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 3415	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
2	1	a1d 22	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝐴 ≠ ∅))
3	2	rexlimiv 2577	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2136   ≠ wne 2336  ∃wrex 2445  ∅c0 3409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-nul 3410

This theorem is referenced by: (None)