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Theorem nnullss 5467
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 4353 . 2 (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦𝐴)
2 vex 3484 . . . . 5 𝑦 ∈ V
32snss 4785 . . . 4 (𝑦𝐴 ↔ {𝑦} ⊆ 𝐴)
42snnz 4776 . . . . 5 {𝑦} ≠ ∅
5 vsnex 5434 . . . . . 6 {𝑦} ∈ V
6 sseq1 4009 . . . . . . 7 (𝑥 = {𝑦} → (𝑥𝐴 ↔ {𝑦} ⊆ 𝐴))
7 neeq1 3003 . . . . . . 7 (𝑥 = {𝑦} → (𝑥 ≠ ∅ ↔ {𝑦} ≠ ∅))
86, 7anbi12d 632 . . . . . 6 (𝑥 = {𝑦} → ((𝑥𝐴𝑥 ≠ ∅) ↔ ({𝑦} ⊆ 𝐴 ∧ {𝑦} ≠ ∅)))
95, 8spcev 3606 . . . . 5 (({𝑦} ⊆ 𝐴 ∧ {𝑦} ≠ ∅) → ∃𝑥(𝑥𝐴𝑥 ≠ ∅))
104, 9mpan2 691 . . . 4 ({𝑦} ⊆ 𝐴 → ∃𝑥(𝑥𝐴𝑥 ≠ ∅))
113, 10sylbi 217 . . 3 (𝑦𝐴 → ∃𝑥(𝑥𝐴𝑥 ≠ ∅))
1211exlimiv 1930 . 2 (∃𝑦 𝑦𝐴 → ∃𝑥(𝑥𝐴𝑥 ≠ ∅))
131, 12sylbi 217 1 (𝐴 ≠ ∅ → ∃𝑥(𝑥𝐴𝑥 ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wex 1779  wcel 2108  wne 2940  wss 3951  c0 4333  {csn 4626
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 2708  ax-sep 5296  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-sn 4627  df-pr 4629
This theorem is referenced by: (None)
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