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Mirrors > Home > MPE Home > Th. List > pssne | Structured version Visualization version GIF version |
Description: Two classes in a proper subclass relationship are not equal. (Contributed by NM, 16-Feb-2015.) |
Ref | Expression |
---|---|
pssne | ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pss 3900 | . 2 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵)) | |
2 | 1 | simprbi 500 | 1 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ≠ wne 2987 ⊆ wss 3881 ⊊ wpss 3882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 210 df-an 400 df-pss 3900 |
This theorem is referenced by: pssned 4026 canthp1lem2 10064 mrissmrcd 16903 xppss12 39409 |
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