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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 2689 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) | |
2 | 1 | biimpcd 239 | . 2 ⊢ (𝐴 ∈ 𝐶 → (𝐴 = 𝐵 → 𝐵 ∈ 𝐶)) |
3 | 2 | con3dimp 457 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → ¬ 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 df-cleq 2615 df-clel 2618 |
This theorem is referenced by: onfununi 7438 suc11reg 8516 cantnfp1lem3 8577 oemapvali 8581 xrge0neqmnf 12276 mreexmrid 16303 supxrnemnf 29534 onint1 32448 maxidln0 33844 rencldnfilem 37384 climlimsupcex 40001 icccncfext 40100 |
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