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Mirrors > Home > MPE Home > Th. List > nssne2 | Structured version Visualization version GIF version |
Description: Two classes are different if they are not subclasses of the same class. (Contributed by NM, 23-Apr-2015.) |
Ref | Expression |
---|---|
nssne2 | ⊢ ((𝐴 ⊆ 𝐶 ∧ ¬ 𝐵 ⊆ 𝐶) → 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1 4021 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)) | |
2 | 1 | biimpcd 249 | . . 3 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 = 𝐵 → 𝐵 ⊆ 𝐶)) |
3 | 2 | necon3bd 2952 | . 2 ⊢ (𝐴 ⊆ 𝐶 → (¬ 𝐵 ⊆ 𝐶 → 𝐴 ≠ 𝐵)) |
4 | 3 | imp 406 | 1 ⊢ ((𝐴 ⊆ 𝐶 ∧ ¬ 𝐵 ⊆ 𝐶) → 𝐴 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ≠ wne 2938 ⊆ wss 3963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-cleq 2727 df-ne 2939 df-ss 3980 |
This theorem is referenced by: atcvatlem 32414 mdsymlem3 32434 disjdifprg 32595 mapdh6aN 41718 mapdh8e 41767 hdmap1l6a 41792 |
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