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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 2713 | . . 3 ⊢ ((𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴) → ¬ 𝐵 = 𝐴) | |
2 | eqcom 2617 | . . 3 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵) | |
3 | 1, 2 | sylnib 317 | . 2 ⊢ ((𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴) → ¬ 𝐴 = 𝐵) |
4 | dfpss2 3654 | . . 3 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵)) | |
5 | 4 | baibr 943 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (¬ 𝐴 = 𝐵 ↔ 𝐴 ⊊ 𝐵)) |
6 | 3, 5 | syl5ib 233 | 1 ⊢ (𝐴 ⊆ 𝐵 → ((𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴) → 𝐴 ⊊ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ⊊ wpss 3541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-cleq 2603 df-clel 2606 df-ne 2782 df-pss 3556 |
This theorem is referenced by: ssnelpssd 3681 ssexnelpss 3682 canthp1lem2 9354 nqpr 9715 uzindi 12643 nthruc 14819 nthruz 14820 vitali 23188 onpsstopbas 31599 |
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