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Mirrors > Home > MPE Home > Th. List > nelneq | Structured version Visualization version GIF version |
Description: A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.) |
Ref | Expression |
---|---|
nelneq | ⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → ¬ 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2866 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) | |
2 | 1 | biimpcd 241 | . 2 ⊢ (𝐴 ∈ 𝐶 → (𝐴 = 𝐵 → 𝐵 ∈ 𝐶)) |
3 | 2 | con3dimp 398 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → ¬ 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-ex 1876 df-cleq 2792 df-clel 2795 |
This theorem is referenced by: onfununi 7677 suc11reg 8766 cantnfp1lem3 8827 oemapvali 8831 xrge0neqmnfOLD 12527 mreexmrid 16618 supxrnemnf 30052 onint1 32956 maxidln0 34331 rencldnfilem 38170 climlimsupcex 40745 icccncfext 40844 |
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