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Mirrors > Home > NFE Home > Th. List > pssdifn0 | 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 3609 | . . . 4 ⊢ (B ⊆ A ↔ (B ∖ A) = ∅) | |
2 | eqss 3287 | . . . . 5 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A)) | |
3 | 2 | simplbi2 608 | . . . 4 ⊢ (A ⊆ B → (B ⊆ A → A = B)) |
4 | 1, 3 | syl5bir 209 | . . 3 ⊢ (A ⊆ B → ((B ∖ A) = ∅ → A = B)) |
5 | 4 | necon3d 2554 | . 2 ⊢ (A ⊆ B → (A ≠ B → (B ∖ A) ≠ ∅)) |
6 | 5 | imp 418 | 1 ⊢ ((A ⊆ B ∧ A ≠ B) → (B ∖ A) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 = wceq 1642 ≠ wne 2516 ∖ cdif 3206 ⊆ wss 3257 ∅c0 3550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-ss 3259 df-nul 3551 |
This theorem is referenced by: pssdif 3612 |
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