Theorem sscon34 3662

Description: Contraposition law for subset. (Contributed by SF, 11-Mar-2015.)

Assertion

Ref	Expression
sscon34	⊢ (A ⊆ B ↔ ∼ B ⊆ ∼ A)

Proof of Theorem sscon34

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		con34b 283	. . . 4 ⊢ ((x ∈ A → x ∈ B) ↔ (¬ x ∈ B → ¬ x ∈ A))
2		vex 2863	. . . . . 6 ⊢ x ∈ V
3	2	elcompl 3226	. . . . 5 ⊢ (x ∈ ∼ B ↔ ¬ x ∈ B)
4	2	elcompl 3226	. . . . 5 ⊢ (x ∈ ∼ A ↔ ¬ x ∈ A)
5	3, 4	imbi12i 316	. . . 4 ⊢ ((x ∈ ∼ B → x ∈ ∼ A) ↔ (¬ x ∈ B → ¬ x ∈ A))
6	1, 5	bitr4i 243	. . 3 ⊢ ((x ∈ A → x ∈ B) ↔ (x ∈ ∼ B → x ∈ ∼ A))
7	6	albii 1566	. 2 ⊢ (∀x(x ∈ A → x ∈ B) ↔ ∀x(x ∈ ∼ B → x ∈ ∼ A))
8		dfss2 3263	. 2 ⊢ (A ⊆ B ↔ ∀x(x ∈ A → x ∈ B))
9		dfss2 3263	. 2 ⊢ ( ∼ B ⊆ ∼ A ↔ ∀x(x ∈ ∼ B → x ∈ ∼ A))
10	7, 8, 9	3bitr4i 268	1 ⊢ (A ⊆ B ↔ ∼ B ⊆ ∼ A)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540   ∈ wcel 1710   ∼ ccompl 3206   ⊆ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by:  sbthlem1  6204