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| Description: Negation of subclass expressed in terms of intersection and proper subclass. (Contributed by NM, 30-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) | 
| Ref | Expression | 
|---|---|
| nssinpss | ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) ⊊ 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | inss1 4237 | . . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
| 2 | 1 | biantrur 530 | . 2 ⊢ ((𝐴 ∩ 𝐵) ≠ 𝐴 ↔ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ≠ 𝐴)) | 
| 3 | dfss2 3969 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴) | |
| 4 | 3 | necon3bbii 2988 | . 2 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) ≠ 𝐴) | 
| 5 | df-pss 3971 | . 2 ⊢ ((𝐴 ∩ 𝐵) ⊊ 𝐴 ↔ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ≠ 𝐴)) | |
| 6 | 2, 4, 5 | 3bitr4i 303 | 1 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) ⊊ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ≠ wne 2940 ∩ cin 3950 ⊆ wss 3951 ⊊ wpss 3952 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-in 3958 df-ss 3968 df-pss 3971 | 
| This theorem is referenced by: fbfinnfr 23849 chrelat2i 32384 | 
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