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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 9640 | . 2 ⊢ ¬ 𝐵 ∈ 𝐵 | |
2 | nelelne 3031 | . 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 2099 ≠ wne 2930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-pr 5433 ax-reg 9635 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1537 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rex 3061 df-v 3464 df-un 3952 df-sn 4634 df-pr 4636 |
This theorem is referenced by: nelaneq 9642 preleqg 9658 dfac2b 10173 disjressuc2 38086 oaomoencom 42983 oenassex 42984 tfsconcat0b 43012 |
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