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Theorem pssdifcom1 4190
Description: Two ways to express overlapping subsets. (Contributed by Stefan O'Rear, 31-Oct-2014.)
Assertion
Ref Expression
pssdifcom1 ((𝐴𝐶𝐵𝐶) → ((𝐶𝐴) ⊊ 𝐵 ↔ (𝐶𝐵) ⊊ 𝐴))

Proof of Theorem pssdifcom1
StepHypRef Expression
1 difcom 4189 . . . 4 ((𝐶𝐴) ⊆ 𝐵 ↔ (𝐶𝐵) ⊆ 𝐴)
21a1i 11 . . 3 ((𝐴𝐶𝐵𝐶) → ((𝐶𝐴) ⊆ 𝐵 ↔ (𝐶𝐵) ⊆ 𝐴))
3 ssconb 3878 . . . . 5 ((𝐵𝐶𝐴𝐶) → (𝐵 ⊆ (𝐶𝐴) ↔ 𝐴 ⊆ (𝐶𝐵)))
43ancoms 468 . . . 4 ((𝐴𝐶𝐵𝐶) → (𝐵 ⊆ (𝐶𝐴) ↔ 𝐴 ⊆ (𝐶𝐵)))
54notbid 307 . . 3 ((𝐴𝐶𝐵𝐶) → (¬ 𝐵 ⊆ (𝐶𝐴) ↔ ¬ 𝐴 ⊆ (𝐶𝐵)))
62, 5anbi12d 749 . 2 ((𝐴𝐶𝐵𝐶) → (((𝐶𝐴) ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ (𝐶𝐴)) ↔ ((𝐶𝐵) ⊆ 𝐴 ∧ ¬ 𝐴 ⊆ (𝐶𝐵))))
7 dfpss3 3827 . 2 ((𝐶𝐴) ⊊ 𝐵 ↔ ((𝐶𝐴) ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ (𝐶𝐴)))
8 dfpss3 3827 . 2 ((𝐶𝐵) ⊊ 𝐴 ↔ ((𝐶𝐵) ⊆ 𝐴 ∧ ¬ 𝐴 ⊆ (𝐶𝐵)))
96, 7, 83bitr4g 303 1 ((𝐴𝐶𝐵𝐶) → ((𝐶𝐴) ⊊ 𝐵 ↔ (𝐶𝐵) ⊊ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  cdif 3704  wss 3707  wpss 3708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723
This theorem is referenced by:  isfin2-2  9325  compssiso  9380
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