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Mirrors > Home > MPE Home > Th. List > pssdifn0 | Structured version Visualization version GIF version |
Description: A proper subclass has a nonempty difference. (Contributed by NM, 3-May-1994.) |
Ref | Expression |
---|---|
pssdifn0 | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵) → (𝐵 ∖ 𝐴) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssdif0 4140 | . . . 4 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∖ 𝐴) = ∅) | |
2 | eqss 3811 | . . . . 5 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
3 | 2 | simplbi2 495 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ 𝐴 → 𝐴 = 𝐵)) |
4 | 1, 3 | syl5bir 235 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝐵 ∖ 𝐴) = ∅ → 𝐴 = 𝐵)) |
5 | 4 | necon3d 2990 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ≠ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅)) |
6 | 5 | imp 396 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵) → (𝐵 ∖ 𝐴) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ≠ wne 2969 ∖ cdif 3764 ⊆ wss 3767 ∅c0 4113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2375 ax-ext 2775 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-v 3385 df-dif 3770 df-in 3774 df-ss 3781 df-nul 4114 |
This theorem is referenced by: pssdif 4143 tz7.7 5965 domdifsn 8283 inf3lem3 8775 isf32lem6 9466 fclscf 22153 flimfnfcls 22156 lebnumlem1 23084 lebnumlem2 23085 lebnumlem3 23086 ig1peu 24268 ig1pdvds 24273 divrngidl 34305 |
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