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Mirrors > Home > MPE Home > Th. List > npss | Structured version Visualization version GIF version |
Description: A class is not a proper subclass of another iff it satisfies a one-directional form of eqss 3960. (Contributed by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
npss | ⊢ (¬ 𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 → 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm4.61 406 | . . 3 ⊢ (¬ (𝐴 ⊆ 𝐵 → 𝐴 = 𝐵) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵)) | |
2 | dfpss2 4046 | . . 3 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵)) | |
3 | 1, 2 | bitr4i 278 | . 2 ⊢ (¬ (𝐴 ⊆ 𝐵 → 𝐴 = 𝐵) ↔ 𝐴 ⊊ 𝐵) |
4 | 3 | con1bii 357 | 1 ⊢ (¬ 𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 → 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ⊆ wss 3911 ⊊ wpss 3912 |
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-ne 2941 df-pss 3930 |
This theorem is referenced by: ttukeylem7 10456 canthp1lem2 10594 pgpfac1lem1 19858 lspsncv0 20623 obslbs 21152 ssmxidl 32287 fvineqsneq 35929 |
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