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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 8778 | . 2 ⊢ ¬ 𝐵 ∈ 𝐵 | |
2 | nelelne 3097 | . 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 2164 ≠ wne 2999 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 ax-reg 8773 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-v 3416 df-dif 3801 df-un 3803 df-nul 4147 df-sn 4400 df-pr 4402 |
This theorem is referenced by: nelaneq 8780 preleqg 8794 dfac2b 9273 |
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