Theorem nsspssun 4188

Description: Negation of subclass expressed in terms of proper subclass and union. (Contributed by NM, 15-Sep-2004.)

Assertion

Ref	Expression
nsspssun	⊢ (¬ 𝐴 ⊆ 𝐵 ↔ 𝐵 ⊊ (𝐴 ∪ 𝐵))

Proof of Theorem nsspssun

Step	Hyp	Ref	Expression
1		ssun2 4103	. . . 4 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
2	1	biantrur 530	. . 3 ⊢ (¬ (𝐴 ∪ 𝐵) ⊆ 𝐵 ↔ (𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ ¬ (𝐴 ∪ 𝐵) ⊆ 𝐵))
3		ssid 3939	. . . . 5 ⊢ 𝐵 ⊆ 𝐵
4	3	biantru 529	. . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐵))
5		unss 4114	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐵) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐵)
6	4, 5	bitri 274	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) ⊆ 𝐵)
7	2, 6	xchnxbir 332	. 2 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ ¬ (𝐴 ∪ 𝐵) ⊆ 𝐵))
8		dfpss3 4017	. 2 ⊢ (𝐵 ⊊ (𝐴 ∪ 𝐵) ↔ (𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ ¬ (𝐴 ∪ 𝐵) ⊆ 𝐵))
9	7, 8	bitr4i 277	1 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ 𝐵 ⊊ (𝐴 ∪ 𝐵))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 395   ∪ cun 3881   ⊆ wss 3883   ⊊ wpss 3884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-v 3424  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902

This theorem is referenced by:  disjpss  4391  lindsenlbs  35699