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Theorem pssdifcom1 3635
 Description: Two ways to express overlapping subsets. (Contributed by Stefan O'Rear, 31-Oct-2014.)
Assertion
Ref Expression
pssdifcom1 ((A C B C) → ((C A) ⊊ B ↔ (C B) ⊊ A))

Proof of Theorem pssdifcom1
StepHypRef Expression
1 difcom 3634 . . . 4 ((C A) B ↔ (C B) A)
21a1i 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)))
43ancoms 439 . . . 4 ((A C B C) → (B (C A) ↔ A (C B)))
54notbid 285 . . 3 ((A C B C) → (¬ B (C A) ↔ ¬ A (C B)))
62, 5anbi12d 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)))
96, 7, 83bitr4g 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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