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| Mirrors > Home > MPE Home > Th. List > ssnelpss | Structured version Visualization version GIF version | ||
| Description: A subclass missing a member is a proper subclass. (Contributed by NM, 12-Jan-2002.) |
| Ref | Expression |
|---|---|
| ssnelpss | ⊢ (𝐴 ⊆ 𝐵 → ((𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴) → 𝐴 ⊊ 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nelneq2 2866 | . . 3 ⊢ ((𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴) → ¬ 𝐵 = 𝐴) | |
| 2 | eqcom 2744 | . . 3 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵) | |
| 3 | 1, 2 | sylnib 328 | . 2 ⊢ ((𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴) → ¬ 𝐴 = 𝐵) |
| 4 | dfpss2 4088 | . . 3 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵)) | |
| 5 | 4 | baibr 536 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (¬ 𝐴 = 𝐵 ↔ 𝐴 ⊊ 𝐵)) |
| 6 | 3, 5 | imbitrid 244 | 1 ⊢ (𝐴 ⊆ 𝐵 → ((𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴) → 𝐴 ⊊ 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ 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-ex 1780 df-cleq 2729 df-clel 2816 df-ne 2941 df-pss 3971 |
| This theorem is referenced by: ssnelpssd 4115 ssexnelpss 4116 isfin4p1 10355 canthp1lem2 10693 nqpr 11054 uzindi 14023 nthruc 16288 nthruz 16289 vitali 25648 onpsstopbas 36431 |
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