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Theorem ssconb 3076
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 2939 . . . . . . 7 (𝐴𝐶 → (𝑥𝐴𝑥𝐶))
2 ssel 2939 . . . . . . 7 (𝐵𝐶 → (𝑥𝐵𝑥𝐶))
3 pm5.1 533 . . . . . . 7 (((𝑥𝐴𝑥𝐶) ∧ (𝑥𝐵𝑥𝐶)) → ((𝑥𝐴𝑥𝐶) ↔ (𝑥𝐵𝑥𝐶)))
41, 2, 3syl2an 273 . . . . . 6 ((𝐴𝐶𝐵𝐶) → ((𝑥𝐴𝑥𝐶) ↔ (𝑥𝐵𝑥𝐶)))
5 con2b 593 . . . . . . 7 ((𝑥𝐴 → ¬ 𝑥𝐵) ↔ (𝑥𝐵 → ¬ 𝑥𝐴))
65a1i 9 . . . . . 6 ((𝐴𝐶𝐵𝐶) → ((𝑥𝐴 → ¬ 𝑥𝐵) ↔ (𝑥𝐵 → ¬ 𝑥𝐴)))
74, 6anbi12d 442 . . . . 5 ((𝐴𝐶𝐵𝐶) → (((𝑥𝐴𝑥𝐶) ∧ (𝑥𝐴 → ¬ 𝑥𝐵)) ↔ ((𝑥𝐵𝑥𝐶) ∧ (𝑥𝐵 → ¬ 𝑥𝐴))))
8 jcab 535 . . . . 5 ((𝑥𝐴 → (𝑥𝐶 ∧ ¬ 𝑥𝐵)) ↔ ((𝑥𝐴𝑥𝐶) ∧ (𝑥𝐴 → ¬ 𝑥𝐵)))
9 jcab 535 . . . . 5 ((𝑥𝐵 → (𝑥𝐶 ∧ ¬ 𝑥𝐴)) ↔ ((𝑥𝐵𝑥𝐶) ∧ (𝑥𝐵 → ¬ 𝑥𝐴)))
107, 8, 93bitr4g 212 . . . 4 ((𝐴𝐶𝐵𝐶) → ((𝑥𝐴 → (𝑥𝐶 ∧ ¬ 𝑥𝐵)) ↔ (𝑥𝐵 → (𝑥𝐶 ∧ ¬ 𝑥𝐴))))
11 eldif 2927 . . . . 5 (𝑥 ∈ (𝐶𝐵) ↔ (𝑥𝐶 ∧ ¬ 𝑥𝐵))
1211imbi2i 215 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐶𝐵)) ↔ (𝑥𝐴 → (𝑥𝐶 ∧ ¬ 𝑥𝐵)))
13 eldif 2927 . . . . 5 (𝑥 ∈ (𝐶𝐴) ↔ (𝑥𝐶 ∧ ¬ 𝑥𝐴))
1413imbi2i 215 . . . 4 ((𝑥𝐵𝑥 ∈ (𝐶𝐴)) ↔ (𝑥𝐵 → (𝑥𝐶 ∧ ¬ 𝑥𝐴)))
1510, 12, 143bitr4g 212 . . 3 ((𝐴𝐶𝐵𝐶) → ((𝑥𝐴𝑥 ∈ (𝐶𝐵)) ↔ (𝑥𝐵𝑥 ∈ (𝐶𝐴))))
1615albidv 1705 . 2 ((𝐴𝐶𝐵𝐶) → (∀𝑥(𝑥𝐴𝑥 ∈ (𝐶𝐵)) ↔ ∀𝑥(𝑥𝐵𝑥 ∈ (𝐶𝐴))))
17 dfss2 2934 . 2 (𝐴 ⊆ (𝐶𝐵) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐶𝐵)))
18 dfss2 2934 . 2 (𝐵 ⊆ (𝐶𝐴) ↔ ∀𝑥(𝑥𝐵𝑥 ∈ (𝐶𝐴)))
1916, 17, 183bitr4g 212 1 ((𝐴𝐶𝐵𝐶) → (𝐴 ⊆ (𝐶𝐵) ↔ 𝐵 ⊆ (𝐶𝐴)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97  wb 98  wal 1241  wcel 1393  cdif 2914  wss 2917
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931
This theorem is referenced by: (None)
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