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Mirrors > Home > MPE Home > Th. List > neldif | Structured version Visualization version GIF version |
Description: Implication of membership in a class difference. (Contributed by NM, 28-Jun-1994.) |
Ref | Expression |
---|---|
neldif | ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ (𝐵 ∖ 𝐶)) → 𝐴 ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 3957 | . . . 4 ⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶)) | |
2 | 1 | simplbi2 500 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ 𝐶 → 𝐴 ∈ (𝐵 ∖ 𝐶))) |
3 | 2 | con1d 145 | . 2 ⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ (𝐵 ∖ 𝐶) → 𝐴 ∈ 𝐶)) |
4 | 3 | imp 406 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ (𝐵 ∖ 𝐶)) → 𝐴 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∈ wcel 2099 ∖ cdif 3944 |
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-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3473 df-dif 3950 |
This theorem is referenced by: peano5 7899 peano5OLD 7900 boxcutc 8960 etransc 45671 |
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