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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 4249. (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 4249 | . 2 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐵 = ∅) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ¬ 𝐵 = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 ∈ wcel 2111 ∅c0 4243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-dif 3884 df-nul 4244 |
This theorem is referenced by: iin0 5226 snsn0non 6277 tfrlem16 8012 hon0 29576 dmadjrnb 29689 bnj98 32249 prv0 32790 bj-isrvec 34708 dvnprodlem3 42590 |
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