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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 4326. (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 4326 | . 2 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐵 = ∅) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ¬ 𝐵 = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2098 ∅c0 4315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-dif 3944 df-nul 4316 |
This theorem is referenced by: iin0 5351 snsn0non 6480 tfrlem16 8389 hon0 31540 dmadjrnb 31653 bnj98 34396 prv0 34938 dvnprodlem3 45209 |
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