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Theorem ssconb 3116
 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 3003 . . . . . . 7 (𝐴𝐶 → (𝑥𝐴𝑥𝐶))
2 ssel 3003 . . . . . . 7 (𝐵𝐶 → (𝑥𝐵𝑥𝐶))
3 pm5.1 566 . . . . . . 7 (((𝑥𝐴𝑥𝐶) ∧ (𝑥𝐵𝑥𝐶)) → ((𝑥𝐴𝑥𝐶) ↔ (𝑥𝐵𝑥𝐶)))
41, 2, 3syl2an 283 . . . . . 6 ((𝐴𝐶𝐵𝐶) → ((𝑥𝐴𝑥𝐶) ↔ (𝑥𝐵𝑥𝐶)))
5 con2b 626 . . . . . . 7 ((𝑥𝐴 → ¬ 𝑥𝐵) ↔ (𝑥𝐵 → ¬ 𝑥𝐴))
65a1i 9 . . . . . 6 ((𝐴𝐶𝐵𝐶) → ((𝑥𝐴 → ¬ 𝑥𝐵) ↔ (𝑥𝐵 → ¬ 𝑥𝐴)))
74, 6anbi12d 457 . . . . 5 ((𝐴𝐶𝐵𝐶) → (((𝑥𝐴𝑥𝐶) ∧ (𝑥𝐴 → ¬ 𝑥𝐵)) ↔ ((𝑥𝐵𝑥𝐶) ∧ (𝑥𝐵 → ¬ 𝑥𝐴))))
8 jcab 568 . . . . 5 ((𝑥𝐴 → (𝑥𝐶 ∧ ¬ 𝑥𝐵)) ↔ ((𝑥𝐴𝑥𝐶) ∧ (𝑥𝐴 → ¬ 𝑥𝐵)))
9 jcab 568 . . . . 5 ((𝑥𝐵 → (𝑥𝐶 ∧ ¬ 𝑥𝐴)) ↔ ((𝑥𝐵𝑥𝐶) ∧ (𝑥𝐵 → ¬ 𝑥𝐴)))
107, 8, 93bitr4g 221 . . . 4 ((𝐴𝐶𝐵𝐶) → ((𝑥𝐴 → (𝑥𝐶 ∧ ¬ 𝑥𝐵)) ↔ (𝑥𝐵 → (𝑥𝐶 ∧ ¬ 𝑥𝐴))))
11 eldif 2992 . . . . 5 (𝑥 ∈ (𝐶𝐵) ↔ (𝑥𝐶 ∧ ¬ 𝑥𝐵))
1211imbi2i 224 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐶𝐵)) ↔ (𝑥𝐴 → (𝑥𝐶 ∧ ¬ 𝑥𝐵)))
13 eldif 2992 . . . . 5 (𝑥 ∈ (𝐶𝐴) ↔ (𝑥𝐶 ∧ ¬ 𝑥𝐴))
1413imbi2i 224 . . . 4 ((𝑥𝐵𝑥 ∈ (𝐶𝐴)) ↔ (𝑥𝐵 → (𝑥𝐶 ∧ ¬ 𝑥𝐴)))
1510, 12, 143bitr4g 221 . . 3 ((𝐴𝐶𝐵𝐶) → ((𝑥𝐴𝑥 ∈ (𝐶𝐵)) ↔ (𝑥𝐵𝑥 ∈ (𝐶𝐴))))
1615albidv 1747 . 2 ((𝐴𝐶𝐵𝐶) → (∀𝑥(𝑥𝐴𝑥 ∈ (𝐶𝐵)) ↔ ∀𝑥(𝑥𝐵𝑥 ∈ (𝐶𝐴))))
17 dfss2 2998 . 2 (𝐴 ⊆ (𝐶𝐵) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐶𝐵)))
18 dfss2 2998 . 2 (𝐵 ⊆ (𝐶𝐴) ↔ ∀𝑥(𝑥𝐵𝑥 ∈ (𝐶𝐴)))
1916, 17, 183bitr4g 221 1 ((𝐴𝐶𝐵𝐶) → (𝐴 ⊆ (𝐶𝐵) ↔ 𝐵 ⊆ (𝐶𝐴)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103  ∀wal 1283   ∈ wcel 1434   ∖ cdif 2980   ⊆ wss 2983 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2613  df-dif 2985  df-in 2989  df-ss 2996 This theorem is referenced by: (None)
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