Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > pssned | Structured version Visualization version GIF version |
Description: Proper subclasses are unequal. Deduction form of pssne 4037. (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 4037 | . 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 2941 ⊊ wpss 3893 |
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 206 df-an 398 df-pss 3911 |
This theorem is referenced by: omsucne 7759 ackbij1lem15 10032 canthnumlem 10446 canthp1lem2 10451 mrieqv2d 17389 slwpss 19258 topdifinffinlem 35559 lsatssn0 37055 islshpcv 37106 lkrpssN 37216 |
Copyright terms: Public domain | W3C validator |