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Mirrors > Home > NFE Home > Th. List > pssdifcom1 | GIF version |
Description: Two ways to express overlapping subsets. (Contributed by Stefan O'Rear, 31-Oct-2014.) |
Ref | Expression |
---|---|
pssdifcom1 | ⊢ ((A ⊆ C ∧ B ⊆ C) → ((C ∖ A) ⊊ B ↔ (C ∖ B) ⊊ A)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difcom 3634 | . . . 4 ⊢ ((C ∖ A) ⊆ B ↔ (C ∖ B) ⊆ A) | |
2 | 1 | a1i 10 | . . 3 ⊢ ((A ⊆ C ∧ B ⊆ C) → ((C ∖ A) ⊆ B ↔ (C ∖ B) ⊆ A)) |
3 | ssconb 3399 | . . . . 5 ⊢ ((B ⊆ C ∧ A ⊆ C) → (B ⊆ (C ∖ A) ↔ A ⊆ (C ∖ B))) | |
4 | 3 | ancoms 439 | . . . 4 ⊢ ((A ⊆ C ∧ B ⊆ C) → (B ⊆ (C ∖ A) ↔ A ⊆ (C ∖ B))) |
5 | 4 | notbid 285 | . . 3 ⊢ ((A ⊆ C ∧ B ⊆ C) → (¬ B ⊆ (C ∖ A) ↔ ¬ A ⊆ (C ∖ B))) |
6 | 2, 5 | anbi12d 691 | . 2 ⊢ ((A ⊆ C ∧ B ⊆ C) → (((C ∖ A) ⊆ B ∧ ¬ B ⊆ (C ∖ A)) ↔ ((C ∖ B) ⊆ A ∧ ¬ A ⊆ (C ∖ B)))) |
7 | dfpss3 3355 | . 2 ⊢ ((C ∖ A) ⊊ B ↔ ((C ∖ A) ⊆ B ∧ ¬ B ⊆ (C ∖ A))) | |
8 | dfpss3 3355 | . 2 ⊢ ((C ∖ B) ⊊ A ↔ ((C ∖ B) ⊆ A ∧ ¬ A ⊆ (C ∖ B))) | |
9 | 6, 7, 8 | 3bitr4g 279 | 1 ⊢ ((A ⊆ C ∧ B ⊆ C) → ((C ∖ A) ⊊ B ↔ (C ∖ B) ⊊ A)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 176 ∧ wa 358 ∖ cdif 3206 ⊆ wss 3257 ⊊ wpss 3258 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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-un 3214 df-dif 3215 df-ss 3259 df-pss 3261 |
This theorem is referenced by: (None) |
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