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Theorem pssdifcom2 4088
Description: Two ways to express non-covering pairs of subsets. (Contributed by Stefan O'Rear, 31-Oct-2014.)
Assertion
Ref Expression
pssdifcom2 ((𝐴𝐶𝐵𝐶) → (𝐵 ⊊ (𝐶𝐴) ↔ 𝐴 ⊊ (𝐶𝐵)))

Proof of Theorem pssdifcom2
StepHypRef Expression
1 ssconb 3776 . . . 4 ((𝐵𝐶𝐴𝐶) → (𝐵 ⊆ (𝐶𝐴) ↔ 𝐴 ⊆ (𝐶𝐵)))
21ancoms 468 . . 3 ((𝐴𝐶𝐵𝐶) → (𝐵 ⊆ (𝐶𝐴) ↔ 𝐴 ⊆ (𝐶𝐵)))
3 difcom 4086 . . . . 5 ((𝐶𝐴) ⊆ 𝐵 ↔ (𝐶𝐵) ⊆ 𝐴)
43notbii 309 . . . 4 (¬ (𝐶𝐴) ⊆ 𝐵 ↔ ¬ (𝐶𝐵) ⊆ 𝐴)
54a1i 11 . . 3 ((𝐴𝐶𝐵𝐶) → (¬ (𝐶𝐴) ⊆ 𝐵 ↔ ¬ (𝐶𝐵) ⊆ 𝐴))
62, 5anbi12d 747 . 2 ((𝐴𝐶𝐵𝐶) → ((𝐵 ⊆ (𝐶𝐴) ∧ ¬ (𝐶𝐴) ⊆ 𝐵) ↔ (𝐴 ⊆ (𝐶𝐵) ∧ ¬ (𝐶𝐵) ⊆ 𝐴)))
7 dfpss3 3726 . 2 (𝐵 ⊊ (𝐶𝐴) ↔ (𝐵 ⊆ (𝐶𝐴) ∧ ¬ (𝐶𝐴) ⊆ 𝐵))
8 dfpss3 3726 . 2 (𝐴 ⊊ (𝐶𝐵) ↔ (𝐴 ⊆ (𝐶𝐵) ∧ ¬ (𝐶𝐵) ⊆ 𝐴))
96, 7, 83bitr4g 303 1 ((𝐴𝐶𝐵𝐶) → (𝐵 ⊊ (𝐶𝐴) ↔ 𝐴 ⊊ (𝐶𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  cdif 3604  wss 3607  wpss 3608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623
This theorem is referenced by:  fin2i2  9178
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