Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssconb Structured version   Visualization version   GIF version

Theorem ssconb 3776
 Description: Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)
Assertion
Ref Expression
ssconb ((𝐴𝐶𝐵𝐶) → (𝐴 ⊆ (𝐶𝐵) ↔ 𝐵 ⊆ (𝐶𝐴)))

Proof of Theorem ssconb
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ssel 3630 . . . . . . 7 (𝐴𝐶 → (𝑥𝐴𝑥𝐶))
2 ssel 3630 . . . . . . 7 (𝐵𝐶 → (𝑥𝐵𝑥𝐶))
3 pm5.1 920 . . . . . . 7 (((𝑥𝐴𝑥𝐶) ∧ (𝑥𝐵𝑥𝐶)) → ((𝑥𝐴𝑥𝐶) ↔ (𝑥𝐵𝑥𝐶)))
41, 2, 3syl2an 493 . . . . . 6 ((𝐴𝐶𝐵𝐶) → ((𝑥𝐴𝑥𝐶) ↔ (𝑥𝐵𝑥𝐶)))
5 con2b 348 . . . . . . 7 ((𝑥𝐴 → ¬ 𝑥𝐵) ↔ (𝑥𝐵 → ¬ 𝑥𝐴))
65a1i 11 . . . . . 6 ((𝐴𝐶𝐵𝐶) → ((𝑥𝐴 → ¬ 𝑥𝐵) ↔ (𝑥𝐵 → ¬ 𝑥𝐴)))
74, 6anbi12d 747 . . . . 5 ((𝐴𝐶𝐵𝐶) → (((𝑥𝐴𝑥𝐶) ∧ (𝑥𝐴 → ¬ 𝑥𝐵)) ↔ ((𝑥𝐵𝑥𝐶) ∧ (𝑥𝐵 → ¬ 𝑥𝐴))))
8 jcab 925 . . . . 5 ((𝑥𝐴 → (𝑥𝐶 ∧ ¬ 𝑥𝐵)) ↔ ((𝑥𝐴𝑥𝐶) ∧ (𝑥𝐴 → ¬ 𝑥𝐵)))
9 jcab 925 . . . . 5 ((𝑥𝐵 → (𝑥𝐶 ∧ ¬ 𝑥𝐴)) ↔ ((𝑥𝐵𝑥𝐶) ∧ (𝑥𝐵 → ¬ 𝑥𝐴)))
107, 8, 93bitr4g 303 . . . 4 ((𝐴𝐶𝐵𝐶) → ((𝑥𝐴 → (𝑥𝐶 ∧ ¬ 𝑥𝐵)) ↔ (𝑥𝐵 → (𝑥𝐶 ∧ ¬ 𝑥𝐴))))
11 eldif 3617 . . . . 5 (𝑥 ∈ (𝐶𝐵) ↔ (𝑥𝐶 ∧ ¬ 𝑥𝐵))
1211imbi2i 325 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐶𝐵)) ↔ (𝑥𝐴 → (𝑥𝐶 ∧ ¬ 𝑥𝐵)))
13 eldif 3617 . . . . 5 (𝑥 ∈ (𝐶𝐴) ↔ (𝑥𝐶 ∧ ¬ 𝑥𝐴))
1413imbi2i 325 . . . 4 ((𝑥𝐵𝑥 ∈ (𝐶𝐴)) ↔ (𝑥𝐵 → (𝑥𝐶 ∧ ¬ 𝑥𝐴)))
1510, 12, 143bitr4g 303 . . 3 ((𝐴𝐶𝐵𝐶) → ((𝑥𝐴𝑥 ∈ (𝐶𝐵)) ↔ (𝑥𝐵𝑥 ∈ (𝐶𝐴))))
1615albidv 1889 . 2 ((𝐴𝐶𝐵𝐶) → (∀𝑥(𝑥𝐴𝑥 ∈ (𝐶𝐵)) ↔ ∀𝑥(𝑥𝐵𝑥 ∈ (𝐶𝐴))))
17 dfss2 3624 . 2 (𝐴 ⊆ (𝐶𝐵) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐶𝐵)))
18 dfss2 3624 . 2 (𝐵 ⊆ (𝐶𝐴) ↔ ∀𝑥(𝑥𝐵𝑥 ∈ (𝐶𝐴)))
1916, 17, 183bitr4g 303 1 ((𝐴𝐶𝐵𝐶) → (𝐴 ⊆ (𝐶𝐵) ↔ 𝐵 ⊆ (𝐶𝐴)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383  ∀wal 1521   ∈ wcel 2030   ∖ cdif 3604   ⊆ wss 3607 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-v 3233  df-dif 3610  df-in 3614  df-ss 3621 This theorem is referenced by:  pssdifcom1  4087  pssdifcom2  4088  sbthlem1  8111  sbthlem2  8112  rpnnen2lem11  14997  setscom  15950  dpjidcl  18503  clsval2  20902  regsep2  21228  ordtconnlem1  30098  conss2  38964
 Copyright terms: Public domain W3C validator