Theorem dfpss3 1558

Description: Alternate definition of proper subclass.

Assertion

Ref	Expression
dfpss3	⊢ (A ⊂ B ↔ (A ⊆ B ∧ ¬ B ⊆ A))

Proof of Theorem dfpss3

Step	Hyp	Ref	Expression
1		eqss 1516	. . . 4 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A))
2	1	negbii 162	. . 3 ⊢ (¬ A = B ↔ ¬ (A ⊆ B ∧ B ⊆ A))
3	2	anbi2i 367	. 2 ⊢ ((A ⊆ B ∧ ¬ A = B) ↔ (A ⊆ B ∧ ¬ (A ⊆ B ∧ B ⊆ A)))
4		dfpss2 1557	. 2 ⊢ (A ⊂ B ↔ (A ⊆ B ∧ ¬ A = B))
5		anclb 264	. . . 4 ⊢ ((A ⊆ B → B ⊆ A) ↔ (A ⊆ B → (A ⊆ B ∧ B ⊆ A)))
6		iman 205	. . . 4 ⊢ ((A ⊆ B → B ⊆ A) ↔ ¬ (A ⊆ B ∧ ¬ B ⊆ A))
7		iman 205	. . . 4 ⊢ ((A ⊆ B → (A ⊆ B ∧ B ⊆ A)) ↔ ¬ (A ⊆ B ∧ ¬ (A ⊆ B ∧ B ⊆ A)))
8	5, 6, 7	3bitr3 156	. . 3 ⊢ (¬ (A ⊆ B ∧ ¬ B ⊆ A) ↔ ¬ (A ⊆ B ∧ ¬ (A ⊆ B ∧ B ⊆ A)))
9	8	bicon4i 401	. 2 ⊢ ((A ⊆ B ∧ ¬ B ⊆ A) ↔ (A ⊆ B ∧ ¬ (A ⊆ B ∧ B ⊆ A)))
10	3, 4, 9	3bitr4 158	1 ⊢ (A ⊂ B ↔ (A ⊆ B ∧ ¬ B ⊆ A))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ⊆ wss 1487   ⊂ wpss 1488

This theorem is referenced by:  pssirr 1570  pssn2lp 1571  nssinpss 1665  nsspssun 1666  php3 3411  prlem934 3933  reclem2pr 3951  ch0psst 5370  chpsscon3t 5420  chpssat 5756

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-in 1491  df-ss 1492  df-pss 1494

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