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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 4334. (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 4334 | . 2 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐵 = ∅) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ¬ 𝐵 = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1534 ∈ wcel 2099 ∅c0 4323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-dif 3950 df-nul 4324 |
This theorem is referenced by: iin0 5362 snsn0non 6494 tfrlem16 8413 hon0 31602 dmadjrnb 31715 bnj98 34498 prv0 35040 dvnprodlem3 45336 |
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