![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > n0ii | Structured version Visualization version GIF version |
Description: If a class has elements, then it is not empty. Inference associated with n0i 4333. (Contributed by BJ, 15-Jul-2021.) |
Ref | Expression |
---|---|
n0ii.1 | ⊢ 𝐴 ∈ 𝐵 |
Ref | Expression |
---|---|
n0ii | ⊢ ¬ 𝐵 = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0ii.1 | . 2 ⊢ 𝐴 ∈ 𝐵 | |
2 | n0i 4333 | . 2 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐵 = ∅) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ¬ 𝐵 = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 ∅c0 4322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-dif 3951 df-nul 4323 |
This theorem is referenced by: iin0 5360 snsn0non 6487 tfrlem16 8390 hon0 31034 dmadjrnb 31147 bnj98 33867 prv0 34410 dvnprodlem3 44651 |
Copyright terms: Public domain | W3C validator |