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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 2883 | . . 3 ⊢ ((𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴) → ¬ 𝐵 = 𝐴) | |
2 | eqcom 2784 | . . 3 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵) | |
3 | 1, 2 | sylnib 320 | . 2 ⊢ ((𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴) → ¬ 𝐴 = 𝐵) |
4 | dfpss2 3913 | . . 3 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵)) | |
5 | 4 | baibr 532 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (¬ 𝐴 = 𝐵 ↔ 𝐴 ⊊ 𝐵)) |
6 | 3, 5 | syl5ib 236 | 1 ⊢ (𝐴 ⊆ 𝐵 → ((𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴) → 𝐴 ⊊ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ⊆ wss 3791 ⊊ wpss 3792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-ext 2753 |
This theorem depends on definitions: df-bi 199 df-an 387 df-ex 1824 df-cleq 2769 df-clel 2773 df-ne 2969 df-pss 3807 |
This theorem is referenced by: ssnelpssd 3940 ssexnelpss 3941 canthp1lem2 9810 nqpr 10171 uzindi 13100 nthruc 15385 nthruz 15386 vitali 23817 onpsstopbas 33012 |
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