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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 3084 | . 2 ⊢ ((𝐶 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐵) → 𝐶 ≠ 𝐴) | |
2 | 1 | expcom 417 | 1 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐵 → 𝐶 ≠ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2111 ≠ wne 2987 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-cleq 2791 df-clel 2870 df-ne 2988 |
This theorem is referenced by: difsn 4691 elneq 9046 frgrncvvdeqlem7 28090 frgrncvvdeqlem9 28092 prter2 36177 |
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