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Mirrors > Home > MPE Home > Th. List > nnullss | Structured version Visualization version GIF version |
Description: A nonempty class (even if proper) has a nonempty subset. (Contributed by NM, 23-Aug-2003.) |
Ref | Expression |
---|---|
nnullss | ⊢ (𝐴 ≠ ∅ → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 4307 | . 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐴) | |
2 | vex 3495 | . . . . 5 ⊢ 𝑦 ∈ V | |
3 | 2 | snss 4710 | . . . 4 ⊢ (𝑦 ∈ 𝐴 ↔ {𝑦} ⊆ 𝐴) |
4 | 2 | snnz 4703 | . . . . 5 ⊢ {𝑦} ≠ ∅ |
5 | snex 5322 | . . . . . 6 ⊢ {𝑦} ∈ V | |
6 | sseq1 3989 | . . . . . . 7 ⊢ (𝑥 = {𝑦} → (𝑥 ⊆ 𝐴 ↔ {𝑦} ⊆ 𝐴)) | |
7 | neeq1 3075 | . . . . . . 7 ⊢ (𝑥 = {𝑦} → (𝑥 ≠ ∅ ↔ {𝑦} ≠ ∅)) | |
8 | 6, 7 | anbi12d 630 | . . . . . 6 ⊢ (𝑥 = {𝑦} → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) ↔ ({𝑦} ⊆ 𝐴 ∧ {𝑦} ≠ ∅))) |
9 | 5, 8 | spcev 3604 | . . . . 5 ⊢ (({𝑦} ⊆ 𝐴 ∧ {𝑦} ≠ ∅) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅)) |
10 | 4, 9 | mpan2 687 | . . . 4 ⊢ ({𝑦} ⊆ 𝐴 → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅)) |
11 | 3, 10 | sylbi 218 | . . 3 ⊢ (𝑦 ∈ 𝐴 → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅)) |
12 | 11 | exlimiv 1922 | . 2 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅)) |
13 | 1, 12 | sylbi 218 | 1 ⊢ (𝐴 ≠ ∅ → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∃wex 1771 ∈ wcel 2105 ≠ wne 3013 ⊆ wss 3933 ∅c0 4288 {csn 4557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-sn 4558 df-pr 4560 |
This theorem is referenced by: (None) |
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