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Mirrors > Home > MPE Home > Th. List > elneq | Structured version Visualization version GIF version |
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 9063 | . 2 ⊢ ¬ 𝐵 ∈ 𝐵 | |
2 | nelelne 3119 | . 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 2114 ≠ wne 3018 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-reg 9058 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-v 3498 df-dif 3941 df-un 3943 df-nul 4294 df-sn 4570 df-pr 4572 |
This theorem is referenced by: nelaneq 9065 preleqg 9080 dfac2b 9558 |
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