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| Description: A class is not equal to any of its elements. (Contributed by AV, 14-Jun-2022.) | 
| Ref | Expression | 
|---|---|
| elneq | ⊢ (𝐴 ∈ 𝐵 → 𝐴 ≠ 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elirr 9637 | . 2 ⊢ ¬ 𝐵 ∈ 𝐵 | |
| 2 | nelelne 3041 | . 2 ⊢ (¬ 𝐵 ∈ 𝐵 → (𝐴 ∈ 𝐵 → 𝐴 ≠ 𝐵)) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ≠ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2108 ≠ wne 2940 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-pr 5432 ax-reg 9632 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-v 3482 df-un 3956 df-sn 4627 df-pr 4629 | 
| This theorem is referenced by: nelaneq 9639 preleqg 9655 dfac2b 10171 disjressuc2 38389 oaomoencom 43330 oenassex 43331 tfsconcat0b 43359 | 
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