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Theorem nnullss 4960
Description: A nonempty class (even if proper) has a nonempty subset. (Contributed by NM, 23-Aug-2003.)
Assertion
Ref Expression
nnullss (𝐴 ≠ ∅ → ∃𝑥(𝑥𝐴𝑥 ≠ ∅))
Distinct variable group:   𝑥,𝐴

Proof of Theorem nnullss
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 n0 3964 . 2 (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦𝐴)
2 vex 3234 . . . . 5 𝑦 ∈ V
32snss 4348 . . . 4 (𝑦𝐴 ↔ {𝑦} ⊆ 𝐴)
42snnz 4340 . . . . 5 {𝑦} ≠ ∅
5 snex 4938 . . . . . 6 {𝑦} ∈ V
6 sseq1 3659 . . . . . . 7 (𝑥 = {𝑦} → (𝑥𝐴 ↔ {𝑦} ⊆ 𝐴))
7 neeq1 2885 . . . . . . 7 (𝑥 = {𝑦} → (𝑥 ≠ ∅ ↔ {𝑦} ≠ ∅))
86, 7anbi12d 747 . . . . . 6 (𝑥 = {𝑦} → ((𝑥𝐴𝑥 ≠ ∅) ↔ ({𝑦} ⊆ 𝐴 ∧ {𝑦} ≠ ∅)))
95, 8spcev 3331 . . . . 5 (({𝑦} ⊆ 𝐴 ∧ {𝑦} ≠ ∅) → ∃𝑥(𝑥𝐴𝑥 ≠ ∅))
104, 9mpan2 707 . . . 4 ({𝑦} ⊆ 𝐴 → ∃𝑥(𝑥𝐴𝑥 ≠ ∅))
113, 10sylbi 207 . . 3 (𝑦𝐴 → ∃𝑥(𝑥𝐴𝑥 ≠ ∅))
1211exlimiv 1898 . 2 (∃𝑦 𝑦𝐴 → ∃𝑥(𝑥𝐴𝑥 ≠ ∅))
131, 12sylbi 207 1 (𝐴 ≠ ∅ → ∃𝑥(𝑥𝐴𝑥 ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wex 1744  wcel 2030  wne 2823  wss 3607  c0 3948  {csn 4210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-sn 4211  df-pr 4213
This theorem is referenced by: (None)
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