Theorem ssconb 3214

Description: Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)

Assertion

Ref	Expression
ssconb	⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 ⊆ (𝐶 ∖ 𝐵) ↔ 𝐵 ⊆ (𝐶 ∖ 𝐴)))

Proof of Theorem ssconb

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3096	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐶 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))
2		ssel 3096	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐶 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶))
3		pm5.1 591	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ∧ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶)) → ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶)))
4	1, 2, 3	syl2an 287	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶)))
5		con2b 659	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴))
6	5	a1i 9	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴)))
7	4, 6	anbi12d 465	. . . . 5 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ∧ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)) ↔ ((𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶) ∧ (𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴))))
8		jcab 593	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵)) ↔ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ∧ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)))
9		jcab 593	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴)) ↔ ((𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶) ∧ (𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴)))
10	7, 8, 9	3bitr4g 222	. . . 4 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → ((𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴))))
11		eldif 3085	. . . . 5 ⊢ (𝑥 ∈ (𝐶 ∖ 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵))
12	11	imbi2i 225	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐶 ∖ 𝐵)) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵)))
13		eldif 3085	. . . . 5 ⊢ (𝑥 ∈ (𝐶 ∖ 𝐴) ↔ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴))
14	13	imbi2i 225	. . . 4 ⊢ ((𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐶 ∖ 𝐴)) ↔ (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴)))
15	10, 12, 14	3bitr4g 222	. . 3 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐶 ∖ 𝐵)) ↔ (𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐶 ∖ 𝐴))))
16	15	albidv 1797	. 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐶 ∖ 𝐵)) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐶 ∖ 𝐴))))
17		dfss2 3091	. 2 ⊢ (𝐴 ⊆ (𝐶 ∖ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐶 ∖ 𝐵)))
18		dfss2 3091	. 2 ⊢ (𝐵 ⊆ (𝐶 ∖ 𝐴) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐶 ∖ 𝐴)))
19	16, 17, 18	3bitr4g 222	1 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 ⊆ (𝐶 ∖ 𝐵) ↔ 𝐵 ⊆ (𝐶 ∖ 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1330   ∈ wcel 1481   ∖ cdif 3073   ⊆ wss 3076

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3078  df-in 3082  df-ss 3089

This theorem is referenced by:  sbthlem1  6853  sbthlem2  6854  setscom  12038