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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 4149. (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 4149 | . 2 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐵 = ∅) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ¬ 𝐵 = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1658 ∈ wcel 2166 ∅c0 4144 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-v 3416 df-dif 3801 df-nul 4145 |
This theorem is referenced by: iin0 5061 snsn0non 6081 tfrlem16 7755 pwcdadom 9353 nnunb 11614 hon0 29207 dmadjrnb 29320 bnj98 31483 dvnprodlem3 40958 |
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