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Mirrors > Home > NFE Home > Th. List > n0i | GIF version |
Description: If a set has elements, it is not empty. (Contributed by NM, 31-Dec-1993.) |
Ref | Expression |
---|---|
n0i | ⊢ (B ∈ A → ¬ A = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3554 | . . 3 ⊢ ¬ B ∈ ∅ | |
2 | eleq2 2414 | . . 3 ⊢ (A = ∅ → (B ∈ A ↔ B ∈ ∅)) | |
3 | 1, 2 | mtbiri 294 | . 2 ⊢ (A = ∅ → ¬ B ∈ A) |
4 | 3 | con2i 112 | 1 ⊢ (B ∈ A → ¬ A = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1642 ∈ wcel 1710 ∅c0 3550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-nul 3551 |
This theorem is referenced by: ne0i 3556 0nelsuc 4400 nndisjeq 4429 nnceleq 4430 sfinltfin 4535 vfin1cltv 4547 funiunfv 5467 ecexr 5950 nceleq 6149 1p1e2c 6155 2p1e3c 6156 |
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