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| Mirrors > Home > MPE Home > Th. List > nelelne | Structured version Visualization version GIF version | ||
| Description: Two classes are different if they don't belong to the same class. (Contributed by Rodolfo Medina, 17-Oct-2010.) (Proof shortened by AV, 10-May-2020.) | 
| Ref | Expression | 
|---|---|
| nelelne | ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐵 → 𝐶 ≠ 𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nelne2 3039 | . 2 ⊢ ((𝐶 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐵) → 𝐶 ≠ 𝐴) | |
| 2 | 1 | expcom 413 | 1 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐵 → 𝐶 ≠ 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2107 ≠ wne 2939 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-cleq 2728 df-clel 2815 df-ne 2940 | 
| This theorem is referenced by: difsn 4797 feldmfvelcdm 7105 resf1extb 7957 elneq 9639 frgrncvvdeqlem7 30325 frgrncvvdeqlem9 30327 prter2 38883 | 
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