Theorem ssconb 3292

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 3173	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐶 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))
2		ssel 3173	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐶 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶))
3		pm5.1 601	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ∧ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶)) → ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶)))
4	1, 2, 3	syl2an 289	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶)))
5		con2b 670	. . . . . . 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 473	. . . . 5 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ∧ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)) ↔ ((𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶) ∧ (𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴))))
8		jcab 603	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵)) ↔ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ∧ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)))
9		jcab 603	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴)) ↔ ((𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶) ∧ (𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴)))
10	7, 8, 9	3bitr4g 223	. . . 4 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → ((𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴))))
11		eldif 3162	. . . . 5 ⊢ (𝑥 ∈ (𝐶 ∖ 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵))
12	11	imbi2i 226	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐶 ∖ 𝐵)) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵)))
13		eldif 3162	. . . . 5 ⊢ (𝑥 ∈ (𝐶 ∖ 𝐴) ↔ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴))
14	13	imbi2i 226	. . . 4 ⊢ ((𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐶 ∖ 𝐴)) ↔ (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴)))
15	10, 12, 14	3bitr4g 223	. . 3 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐶 ∖ 𝐵)) ↔ (𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐶 ∖ 𝐴))))
16	15	albidv 1835	. 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐶 ∖ 𝐵)) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐶 ∖ 𝐴))))
17		dfss2 3168	. 2 ⊢ (𝐴 ⊆ (𝐶 ∖ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐶 ∖ 𝐵)))
18		dfss2 3168	. 2 ⊢ (𝐵 ⊆ (𝐶 ∖ 𝐴) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐶 ∖ 𝐴)))
19	16, 17, 18	3bitr4g 223	1 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 ⊆ (𝐶 ∖ 𝐵) ↔ 𝐵 ⊆ (𝐶 ∖ 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1362   ∈ wcel 2164   ∖ cdif 3150   ⊆ wss 3153

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3155  df-in 3159  df-ss 3166

This theorem is referenced by:  sbthlem1  7016  sbthlem2  7017  setscom  12658