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Theorem pssdifcom1 4006
 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 4005 . . . 4 ((𝐶𝐴) ⊆ 𝐵 ↔ (𝐶𝐵) ⊆ 𝐴)
21a1i 11 . . 3 ((𝐴𝐶𝐵𝐶) → ((𝐶𝐴) ⊆ 𝐵 ↔ (𝐶𝐵) ⊆ 𝐴))
3 ssconb 3705 . . . . 5 ((𝐵𝐶𝐴𝐶) → (𝐵 ⊆ (𝐶𝐴) ↔ 𝐴 ⊆ (𝐶𝐵)))
43ancoms 468 . . . 4 ((𝐴𝐶𝐵𝐶) → (𝐵 ⊆ (𝐶𝐴) ↔ 𝐴 ⊆ (𝐶𝐵)))
54notbid 307 . . 3 ((𝐴𝐶𝐵𝐶) → (¬ 𝐵 ⊆ (𝐶𝐴) ↔ ¬ 𝐴 ⊆ (𝐶𝐵)))
62, 5anbi12d 743 . 2 ((𝐴𝐶𝐵𝐶) → (((𝐶𝐴) ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ (𝐶𝐴)) ↔ ((𝐶𝐵) ⊆ 𝐴 ∧ ¬ 𝐴 ⊆ (𝐶𝐵))))
7 dfpss3 3655 . 2 ((𝐶𝐴) ⊊ 𝐵 ↔ ((𝐶𝐴) ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ (𝐶𝐴)))
8 dfpss3 3655 . 2 ((𝐶𝐵) ⊊ 𝐴 ↔ ((𝐶𝐵) ⊆ 𝐴 ∧ ¬ 𝐴 ⊆ (𝐶𝐵)))
96, 7, 83bitr4g 302 1 ((𝐴𝐶𝐵𝐶) → ((𝐶𝐴) ⊊ 𝐵 ↔ (𝐶𝐵) ⊊ 𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∖ cdif 3537   ⊆ wss 3540   ⊊ wpss 3541 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556 This theorem is referenced by:  isfin2-2  9024  compssiso  9079
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