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Mirrors > Home > MPE Home > Th. List > Mathboxes > nelbrnelim | Structured version Visualization version GIF version |
Description: If a set is related to another set by the negated membership relation, then it is not a member of the other set. (Contributed by AV, 26-Dec-2021.) |
Ref | Expression |
---|---|
nelbrnelim | ⊢ (𝐴 _∉ 𝐵 → 𝐴 ∉ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nelbrim 42990 | . 2 ⊢ (𝐴 _∉ 𝐵 → ¬ 𝐴 ∈ 𝐵) | |
2 | df-nel 3091 | . 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵) | |
3 | 1, 2 | sylibr 235 | 1 ⊢ (𝐴 _∉ 𝐵 → 𝐴 ∉ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2081 ∉ wnel 3090 class class class wbr 4962 _∉ cnelbr 42986 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pr 5221 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-nel 3091 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-br 4963 df-opab 5025 df-xp 5449 df-rel 5450 df-nelbr 42987 |
This theorem is referenced by: (None) |
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