Theorem rexn0 3492

Description: Restricted existential quantification implies its restriction is nonempty (it is also inhabited as shown in rexm 3493). (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 3400	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
2	1	a1d 22	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝐴 ≠ ∅))
3	2	rexlimiv 2568	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2128   ≠ wne 2327  ∃wrex 2436  ∅c0 3394

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-v 2714  df-dif 3104  df-nul 3395

This theorem is referenced by: (None)