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Theorem ssconb 4117
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 3964 . . . . . . 7 (𝐴𝐶 → (𝑥𝐴𝑥𝐶))
2 ssel 3964 . . . . . . 7 (𝐵𝐶 → (𝑥𝐵𝑥𝐶))
3 pm5.1 821 . . . . . . 7 (((𝑥𝐴𝑥𝐶) ∧ (𝑥𝐵𝑥𝐶)) → ((𝑥𝐴𝑥𝐶) ↔ (𝑥𝐵𝑥𝐶)))
41, 2, 3syl2an 597 . . . . . 6 ((𝐴𝐶𝐵𝐶) → ((𝑥𝐴𝑥𝐶) ↔ (𝑥𝐵𝑥𝐶)))
5 con2b 362 . . . . . . 7 ((𝑥𝐴 → ¬ 𝑥𝐵) ↔ (𝑥𝐵 → ¬ 𝑥𝐴))
65a1i 11 . . . . . 6 ((𝐴𝐶𝐵𝐶) → ((𝑥𝐴 → ¬ 𝑥𝐵) ↔ (𝑥𝐵 → ¬ 𝑥𝐴)))
74, 6anbi12d 632 . . . . 5 ((𝐴𝐶𝐵𝐶) → (((𝑥𝐴𝑥𝐶) ∧ (𝑥𝐴 → ¬ 𝑥𝐵)) ↔ ((𝑥𝐵𝑥𝐶) ∧ (𝑥𝐵 → ¬ 𝑥𝐴))))
8 jcab 520 . . . . 5 ((𝑥𝐴 → (𝑥𝐶 ∧ ¬ 𝑥𝐵)) ↔ ((𝑥𝐴𝑥𝐶) ∧ (𝑥𝐴 → ¬ 𝑥𝐵)))
9 jcab 520 . . . . 5 ((𝑥𝐵 → (𝑥𝐶 ∧ ¬ 𝑥𝐴)) ↔ ((𝑥𝐵𝑥𝐶) ∧ (𝑥𝐵 → ¬ 𝑥𝐴)))
107, 8, 93bitr4g 316 . . . 4 ((𝐴𝐶𝐵𝐶) → ((𝑥𝐴 → (𝑥𝐶 ∧ ¬ 𝑥𝐵)) ↔ (𝑥𝐵 → (𝑥𝐶 ∧ ¬ 𝑥𝐴))))
11 eldif 3949 . . . . 5 (𝑥 ∈ (𝐶𝐵) ↔ (𝑥𝐶 ∧ ¬ 𝑥𝐵))
1211imbi2i 338 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐶𝐵)) ↔ (𝑥𝐴 → (𝑥𝐶 ∧ ¬ 𝑥𝐵)))
13 eldif 3949 . . . . 5 (𝑥 ∈ (𝐶𝐴) ↔ (𝑥𝐶 ∧ ¬ 𝑥𝐴))
1413imbi2i 338 . . . 4 ((𝑥𝐵𝑥 ∈ (𝐶𝐴)) ↔ (𝑥𝐵 → (𝑥𝐶 ∧ ¬ 𝑥𝐴)))
1510, 12, 143bitr4g 316 . . 3 ((𝐴𝐶𝐵𝐶) → ((𝑥𝐴𝑥 ∈ (𝐶𝐵)) ↔ (𝑥𝐵𝑥 ∈ (𝐶𝐴))))
1615albidv 1920 . 2 ((𝐴𝐶𝐵𝐶) → (∀𝑥(𝑥𝐴𝑥 ∈ (𝐶𝐵)) ↔ ∀𝑥(𝑥𝐵𝑥 ∈ (𝐶𝐴))))
17 dfss2 3958 . 2 (𝐴 ⊆ (𝐶𝐵) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐶𝐵)))
18 dfss2 3958 . 2 (𝐵 ⊆ (𝐶𝐴) ↔ ∀𝑥(𝑥𝐵𝑥 ∈ (𝐶𝐴)))
1916, 17, 183bitr4g 316 1 ((𝐴𝐶𝐵𝐶) → (𝐴 ⊆ (𝐶𝐵) ↔ 𝐵 ⊆ (𝐶𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wal 1534  wcel 2113  cdif 3936  wss 3939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-v 3499  df-dif 3942  df-in 3946  df-ss 3955
This theorem is referenced by:  pssdifcom1  4438  pssdifcom2  4439  sbthlem1  8630  sbthlem2  8631  rpnnen2lem11  15580  setscom  16530  dpjidcl  19183  clsval2  21661  regsep2  21987  cyc3conja  30803  ordtconnlem1  31171  conss2  40781
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