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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 9635 | . 2 ⊢ ¬ 𝐵 ∈ 𝐵 | |
2 | nelelne 3039 | . 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 2106 ≠ wne 2938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-pr 5438 ax-reg 9630 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-v 3480 df-un 3968 df-sn 4632 df-pr 4634 |
This theorem is referenced by: nelaneq 9637 preleqg 9653 dfac2b 10169 disjressuc2 38370 oaomoencom 43307 oenassex 43308 tfsconcat0b 43336 |
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