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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 3813. (Contributed by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
npss | ⊢ (¬ 𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 → 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm4.61 394 | . . 3 ⊢ (¬ (𝐴 ⊆ 𝐵 → 𝐴 = 𝐵) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵)) | |
2 | dfpss2 3889 | . . 3 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵)) | |
3 | 1, 2 | bitr4i 270 | . 2 ⊢ (¬ (𝐴 ⊆ 𝐵 → 𝐴 = 𝐵) ↔ 𝐴 ⊊ 𝐵) |
4 | 3 | con1bii 348 | 1 ⊢ (¬ 𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 → 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ⊆ wss 3769 ⊊ wpss 3770 |
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 199 df-an 386 df-ne 2972 df-pss 3785 |
This theorem is referenced by: ttukeylem7 9625 canthp1lem2 9763 pgpfac1lem1 18789 lspsncv0 19468 lspsncv0OLD 19469 obslbs 20399 |
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