Theorem brsdom 3286

Description: Strict dominance relation, meaning "B is strictly greater in size than A." Definition of [Mendelson] p. 255.

Assertion

Ref	Expression
brsdom	⊢ (A ≺ B ↔ (A ≼ B ∧ ¬ A ≈ B))

Proof of Theorem brsdom

Step	Hyp	Ref	Expression
1		df-sdom 3276	. . 3 ⊢ ≺ = ( ≼ ∖ ≈ )
2	1	eleq2i 1153	. 2 ⊢ (⟨A, B⟩ ∈ ≺ ↔ ⟨A, B⟩ ∈ ( ≼ ∖ ≈ ))
3		df-br 2063	. 2 ⊢ (A ≺ B ↔ ⟨A, B⟩ ∈ ≺ )
4		df-br 2063	. . . 4 ⊢ (A ≼ B ↔ ⟨A, B⟩ ∈ ≼ )
5		df-br 2063	. . . . 5 ⊢ (A ≈ B ↔ ⟨A, B⟩ ∈ ≈ )
6	5	negbii 162	. . . 4 ⊢ (¬ A ≈ B ↔ ¬ ⟨A, B⟩ ∈ ≈ )
7	4, 6	anbi12i 369	. . 3 ⊢ ((A ≼ B ∧ ¬ A ≈ B) ↔ (⟨A, B⟩ ∈ ≼ ∧ ¬ ⟨A, B⟩ ∈ ≈ ))
8		eldif 1496	. . 3 ⊢ (⟨A, B⟩ ∈ ( ≼ ∖ ≈ ) ↔ (⟨A, B⟩ ∈ ≼ ∧ ¬ ⟨A, B⟩ ∈ ≈ ))
9	7, 8	bitr4 154	. 2 ⊢ ((A ≼ B ∧ ¬ A ≈ B) ↔ ⟨A, B⟩ ∈ ( ≼ ∖ ≈ ))
10	2, 3, 9	3bitr4 158	1 ⊢ (A ≺ B ↔ (A ≼ B ∧ ¬ A ≈ B))

Syntax hints:  ¬ wn 1   ↔ wb 127   ∧ wa 196   ∈ wcel 1092   ∖ cdif 1484  ⟨cop 1810   class class class wbr 2054   ≈ cen 3271   ≼ cdom 3272   ≺ csdm 3273

This theorem is referenced by:  sdomdom 3290  sdomnen 3291  0sdom 3368  ensdomtr 3372  domsdomtr 3374  canth2 3381  php2 3410  php3 3411  nnsdomo 3417  infsdomnn 3426  unfi2 3442  isfinite 3480  nnsdom 3481  cardsdom 3643  cardsdomel 3658  alephordi 3679  alephord 3680  ruc 4924

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-v 1349  df-dif 1489  df-br 2063  df-sdom 3276

metamath.org