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Theorem ssconb 3070
Description: Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)
Assertion
Ref Expression
ssconb ((A𝐶 B𝐶) → (A ⊆ (𝐶B) ↔ B ⊆ (𝐶A)))

Proof of Theorem ssconb
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 ssel 2933 . . . . . . 7 (A𝐶 → (x Ax 𝐶))
2 ssel 2933 . . . . . . 7 (B𝐶 → (x Bx 𝐶))
3 pm5.1 533 . . . . . . 7 (((x Ax 𝐶) (x Bx 𝐶)) → ((x Ax 𝐶) ↔ (x Bx 𝐶)))
41, 2, 3syl2an 273 . . . . . 6 ((A𝐶 B𝐶) → ((x Ax 𝐶) ↔ (x Bx 𝐶)))
5 con2b 592 . . . . . . 7 ((x A → ¬ x B) ↔ (x B → ¬ x A))
65a1i 9 . . . . . 6 ((A𝐶 B𝐶) → ((x A → ¬ x B) ↔ (x B → ¬ x A)))
74, 6anbi12d 442 . . . . 5 ((A𝐶 B𝐶) → (((x Ax 𝐶) (x A → ¬ x B)) ↔ ((x Bx 𝐶) (x B → ¬ x A))))
8 jcab 535 . . . . 5 ((x A → (x 𝐶 ¬ x B)) ↔ ((x Ax 𝐶) (x A → ¬ x B)))
9 jcab 535 . . . . 5 ((x B → (x 𝐶 ¬ x A)) ↔ ((x Bx 𝐶) (x B → ¬ x A)))
107, 8, 93bitr4g 212 . . . 4 ((A𝐶 B𝐶) → ((x A → (x 𝐶 ¬ x B)) ↔ (x B → (x 𝐶 ¬ x A))))
11 eldif 2921 . . . . 5 (x (𝐶B) ↔ (x 𝐶 ¬ x B))
1211imbi2i 215 . . . 4 ((x Ax (𝐶B)) ↔ (x A → (x 𝐶 ¬ x B)))
13 eldif 2921 . . . . 5 (x (𝐶A) ↔ (x 𝐶 ¬ x A))
1413imbi2i 215 . . . 4 ((x Bx (𝐶A)) ↔ (x B → (x 𝐶 ¬ x A)))
1510, 12, 143bitr4g 212 . . 3 ((A𝐶 B𝐶) → ((x Ax (𝐶B)) ↔ (x Bx (𝐶A))))
1615albidv 1702 . 2 ((A𝐶 B𝐶) → (x(x Ax (𝐶B)) ↔ x(x Bx (𝐶A))))
17 dfss2 2928 . 2 (A ⊆ (𝐶B) ↔ x(x Ax (𝐶B)))
18 dfss2 2928 . 2 (B ⊆ (𝐶A) ↔ x(x Bx (𝐶A)))
1916, 17, 183bitr4g 212 1 ((A𝐶 B𝐶) → (A ⊆ (𝐶B) ↔ B ⊆ (𝐶A)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98  wal 1240   wcel 1390  cdif 2908  wss 2911
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925
This theorem is referenced by: (None)
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