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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 3970 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)) | |
2 | 1 | biimpcd 249 | . . 3 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 = 𝐵 → 𝐵 ⊆ 𝐶)) |
3 | 2 | necon3bd 2954 | . 2 ⊢ (𝐴 ⊆ 𝐶 → (¬ 𝐵 ⊆ 𝐶 → 𝐴 ≠ 𝐵)) |
4 | 3 | imp 408 | 1 ⊢ ((𝐴 ⊆ 𝐶 ∧ ¬ 𝐵 ⊆ 𝐶) → 𝐴 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ≠ wne 2940 ⊆ wss 3911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-v 3446 df-in 3918 df-ss 3928 |
This theorem is referenced by: atcvatlem 31369 mdsymlem3 31389 disjdifprg 31539 mapdh6aN 40244 mapdh8e 40293 hdmap1l6a 40318 |
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