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| Mirrors > Home > MPE Home > Th. List > pssned | Structured version Visualization version GIF version | ||
| Description: Proper subclasses are unequal. Deduction form of pssne 4099. (Contributed by David Moews, 1-May-2017.) |
| Ref | Expression |
|---|---|
| pssssd.1 | ⊢ (𝜑 → 𝐴 ⊊ 𝐵) |
| Ref | Expression |
|---|---|
| pssned | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pssssd.1 | . 2 ⊢ (𝜑 → 𝐴 ⊊ 𝐵) | |
| 2 | pssne 4099 | . 2 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ≠ 𝐵) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ≠ wne 2940 ⊊ wpss 3952 |
| 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 207 df-an 396 df-pss 3971 |
| This theorem is referenced by: omsucne 7906 ackbij1lem15 10273 canthnumlem 10688 canthp1lem2 10693 mrieqv2d 17682 slwpss 19630 hashpss 32813 topdifinffinlem 37348 lsatssn0 39003 islshpcv 39054 lkrpssN 39164 |
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