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Theorem ssconb 3240
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 3122 . . . . . . 7 (𝐴𝐶 → (𝑥𝐴𝑥𝐶))
2 ssel 3122 . . . . . . 7 (𝐵𝐶 → (𝑥𝐵𝑥𝐶))
3 pm5.1 591 . . . . . . 7 (((𝑥𝐴𝑥𝐶) ∧ (𝑥𝐵𝑥𝐶)) → ((𝑥𝐴𝑥𝐶) ↔ (𝑥𝐵𝑥𝐶)))
41, 2, 3syl2an 287 . . . . . 6 ((𝐴𝐶𝐵𝐶) → ((𝑥𝐴𝑥𝐶) ↔ (𝑥𝐵𝑥𝐶)))
5 con2b 659 . . . . . . 7 ((𝑥𝐴 → ¬ 𝑥𝐵) ↔ (𝑥𝐵 → ¬ 𝑥𝐴))
65a1i 9 . . . . . 6 ((𝐴𝐶𝐵𝐶) → ((𝑥𝐴 → ¬ 𝑥𝐵) ↔ (𝑥𝐵 → ¬ 𝑥𝐴)))
74, 6anbi12d 465 . . . . 5 ((𝐴𝐶𝐵𝐶) → (((𝑥𝐴𝑥𝐶) ∧ (𝑥𝐴 → ¬ 𝑥𝐵)) ↔ ((𝑥𝐵𝑥𝐶) ∧ (𝑥𝐵 → ¬ 𝑥𝐴))))
8 jcab 593 . . . . 5 ((𝑥𝐴 → (𝑥𝐶 ∧ ¬ 𝑥𝐵)) ↔ ((𝑥𝐴𝑥𝐶) ∧ (𝑥𝐴 → ¬ 𝑥𝐵)))
9 jcab 593 . . . . 5 ((𝑥𝐵 → (𝑥𝐶 ∧ ¬ 𝑥𝐴)) ↔ ((𝑥𝐵𝑥𝐶) ∧ (𝑥𝐵 → ¬ 𝑥𝐴)))
107, 8, 93bitr4g 222 . . . 4 ((𝐴𝐶𝐵𝐶) → ((𝑥𝐴 → (𝑥𝐶 ∧ ¬ 𝑥𝐵)) ↔ (𝑥𝐵 → (𝑥𝐶 ∧ ¬ 𝑥𝐴))))
11 eldif 3111 . . . . 5 (𝑥 ∈ (𝐶𝐵) ↔ (𝑥𝐶 ∧ ¬ 𝑥𝐵))
1211imbi2i 225 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐶𝐵)) ↔ (𝑥𝐴 → (𝑥𝐶 ∧ ¬ 𝑥𝐵)))
13 eldif 3111 . . . . 5 (𝑥 ∈ (𝐶𝐴) ↔ (𝑥𝐶 ∧ ¬ 𝑥𝐴))
1413imbi2i 225 . . . 4 ((𝑥𝐵𝑥 ∈ (𝐶𝐴)) ↔ (𝑥𝐵 → (𝑥𝐶 ∧ ¬ 𝑥𝐴)))
1510, 12, 143bitr4g 222 . . 3 ((𝐴𝐶𝐵𝐶) → ((𝑥𝐴𝑥 ∈ (𝐶𝐵)) ↔ (𝑥𝐵𝑥 ∈ (𝐶𝐴))))
1615albidv 1804 . 2 ((𝐴𝐶𝐵𝐶) → (∀𝑥(𝑥𝐴𝑥 ∈ (𝐶𝐵)) ↔ ∀𝑥(𝑥𝐵𝑥 ∈ (𝐶𝐴))))
17 dfss2 3117 . 2 (𝐴 ⊆ (𝐶𝐵) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐶𝐵)))
18 dfss2 3117 . 2 (𝐵 ⊆ (𝐶𝐴) ↔ ∀𝑥(𝑥𝐵𝑥 ∈ (𝐶𝐴)))
1916, 17, 183bitr4g 222 1 ((𝐴𝐶𝐵𝐶) → (𝐴 ⊆ (𝐶𝐵) ↔ 𝐵 ⊆ (𝐶𝐴)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wal 1333  wcel 2128  cdif 3099  wss 3102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-in 3108  df-ss 3115
This theorem is referenced by:  sbthlem1  6901  sbthlem2  6902  setscom  12241
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