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| 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 4339. (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 4339 | . 2 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐵 = ∅) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ ¬ 𝐵 = ∅ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 = wceq 1539 ∈ wcel 2107 ∅c0 4332 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-dif 3953 df-nul 4333 | 
| This theorem is referenced by: iin0 5361 snsn0non 6508 tfrlem16 8434 hon0 31813 dmadjrnb 31926 bnj98 34882 prv0 35436 dvnprodlem3 45968 | 
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