Theorem nnullss 5437

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 4328	. 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐴)
2		vex 3463	. . . . 5 ⊢ 𝑦 ∈ V
3	2	snss 4761	. . . 4 ⊢ (𝑦 ∈ 𝐴 ↔ {𝑦} ⊆ 𝐴)
4	2	snnz 4752	. . . . 5 ⊢ {𝑦} ≠ ∅
5		vsnex 5404	. . . . . 6 ⊢ {𝑦} ∈ V
6		sseq1 3984	. . . . . . 7 ⊢ (𝑥 = {𝑦} → (𝑥 ⊆ 𝐴 ↔ {𝑦} ⊆ 𝐴))
7		neeq1 2994	. . . . . . 7 ⊢ (𝑥 = {𝑦} → (𝑥 ≠ ∅ ↔ {𝑦} ≠ ∅))
8	6, 7	anbi12d 632	. . . . . 6 ⊢ (𝑥 = {𝑦} → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) ↔ ({𝑦} ⊆ 𝐴 ∧ {𝑦} ≠ ∅)))
9	5, 8	spcev 3585	. . . . 5 ⊢ (({𝑦} ⊆ 𝐴 ∧ {𝑦} ≠ ∅) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅))
10	4, 9	mpan2 691	. . . 4 ⊢ ({𝑦} ⊆ 𝐴 → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅))
11	3, 10	sylbi 217	. . 3 ⊢ (𝑦 ∈ 𝐴 → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅))
12	11	exlimiv 1930	. 2 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅))
13	1, 12	sylbi 217	1 ⊢ (𝐴 ≠ ∅ → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108   ≠ wne 2932   ⊆ wss 3926  ∅c0 4308  {csn 4601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-sn 4602  df-pr 4604

This theorem is referenced by: (None)