Theorem brsdom 8970

Description: Strict dominance relation, meaning "𝐵 is strictly greater in size than 𝐴". Definition of [Mendelson] p. 255. (Contributed by NM, 25-Jun-1998.)

Assertion

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

Proof of Theorem brsdom

Step	Hyp	Ref	Expression
1		df-sdom 8945	. . 3 ⊢ ≺ = ( ≼ ∖ ≈ )
2	1	eleq2i 2861	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ ≺ ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≼ ∖ ≈ ))
3		df-br 5114	. 2 ⊢ (𝐴 ≺ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≺ )
4		df-br 5114	. . . 4 ⊢ (𝐴 ≼ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≼ )
5		df-br 5114	. . . . 5 ⊢ (𝐴 ≈ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≈ )
6	5	notbii 323	. . . 4 ⊢ (¬ 𝐴 ≈ 𝐵 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ ≈ )
7	4, 6	anbi12i 639	. . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ ≼ ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ ≈ ))
8		eldif 3923	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ ( ≼ ∖ ≈ ) ↔ (⟨𝐴, 𝐵⟩ ∈ ≼ ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ ≈ ))
9	7, 8	bitr4i 281	. 2 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵) ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≼ ∖ ≈ ))
10	2, 3, 9	3bitr4i 306	1 ⊢ (𝐴 ≺ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵))

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 400   ∈ wcel 2149   ∖ cdif 3910  ⟨cop 4600   class class class wbr 5113   ≈ cen 8939   ≼ cdom 8940   ≺ csdm 8941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-dif 3916  df-br 5114  df-sdom 8945

This theorem is referenced by:  sdomdom  8976  sdomnen  8977  0sdomg  9093  sdom0  9096  sdomdomtr  9097  domsdomtr  9099  domtriord  9110  canth2  9117  sdomdomtrfi  9184  domsdomtrfi  9185  php2  9191  nnsdomo  9202  1sdom2  9207  sdom1  9209  1sdom2dom  9213  nnsdomg  9258  card2inf  9516  cardsdomelir  9958  cardsdom2  9973  fidomtri2  9979  cardmin2  9984  alephordi  10057  alephord  10058  isfin4p1  10298  isfin5-2  10374  canthnum  10633  canthwe  10635  canthp1  10638  gchdjuidm  10652  gchxpidm  10653  gchhar  10663  axgroth6  10812  hashsdom  14416  ruc  16298  iscard5  44153