Theorem nnullss 5434

Description: A nonempty class (even if proper) has a nonempty subset. (Contributed by NM, 23-Aug-2003.)

Assertion

Ref	Expression
nnullss	⊢ (𝐴 ≠ ∅ → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅))

Distinct variable group:   𝑥,𝐴

Proof of Theorem nnullss

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 4308	. 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐴)
2		vex 3461	. . . . 5 ⊢ 𝑦 ∈ V
3	2	snss 4746	. . . 4 ⊢ (𝑦 ∈ 𝐴 ↔ {𝑦} ⊆ 𝐴)
4	2	snnz 4738	. . . . 5 ⊢ {𝑦} ≠ ∅
5		vsnex 5397	. . . . . 6 ⊢ {𝑦} ∈ V
6		sseq1 3964	. . . . . . 7 ⊢ (𝑥 = {𝑦} → (𝑥 ⊆ 𝐴 ↔ {𝑦} ⊆ 𝐴))
7		neeq1 3022	. . . . . . 7 ⊢ (𝑥 = {𝑦} → (𝑥 ≠ ∅ ↔ {𝑦} ≠ ∅))
8	6, 7	anbi12d 643	. . . . . 6 ⊢ (𝑥 = {𝑦} → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) ↔ ({𝑦} ⊆ 𝐴 ∧ {𝑦} ≠ ∅)))
9	5, 8	spcev 3568	. . . . 5 ⊢ (({𝑦} ⊆ 𝐴 ∧ {𝑦} ≠ ∅) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅))
10	4, 9	mpan2 703	. . . 4 ⊢ ({𝑦} ⊆ 𝐴 → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅))
11	3, 10	sylbi 220	. . 3 ⊢ (𝑦 ∈ 𝐴 → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅))
12	11	exlimiv 1953	. 2 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅))
13	1, 12	sylbi 220	1 ⊢ (𝐴 ≠ ∅ → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563  ∃wex 1802   ∈ wcel 2145   ≠ wne 2960   ⊆ wss 3907  ∅c0 4288  {csn 4585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-sn 4586  df-pr 4588

This theorem is referenced by: (None)