Theorem nsspssun 4217

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 4128	. . . 4 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
2	1	biantrur 540	. . 3 ⊢ (¬ (𝐴 ∪ 𝐵) ⊆ 𝐵 ↔ (𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ ¬ (𝐴 ∪ 𝐵) ⊆ 𝐵))
3		ssid 3956	. . . . 5 ⊢ 𝐵 ⊆ 𝐵
4	3	biantru 539	. . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐵))
5		unss 4139	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐵) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐵)
6	4, 5	bitri 278	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) ⊆ 𝐵)
7	2, 6	xchnxbir 336	. 2 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ ¬ (𝐴 ∪ 𝐵) ⊆ 𝐵))
8		dfpss3 4040	. 2 ⊢ (𝐵 ⊊ (𝐴 ∪ 𝐵) ↔ (𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ ¬ (𝐴 ∪ 𝐵) ⊆ 𝐵))
9	7, 8	bitr4i 281	1 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ 𝐵 ⊊ (𝐴 ∪ 𝐵))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∪ cun 3900   ⊆ wss 3902   ⊊ wpss 3903

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-v 3455  df-un 3907  df-ss 3919  df-pss 3922

This theorem is used by:  disjpss  4417  lindsenlbs  22070