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Theorem brsdom 8146
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 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵))

Proof of Theorem brsdom
StepHypRef Expression
1 df-sdom 8126 . . 3 ≺ = ( ≼ ∖ ≈ )
21eleq2i 2831 . 2 (⟨𝐴, 𝐵⟩ ∈ ≺ ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≼ ∖ ≈ ))
3 df-br 4805 . 2 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≺ )
4 df-br 4805 . . . 4 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≼ )
5 df-br 4805 . . . . 5 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≈ )
65notbii 309 . . . 4 𝐴𝐵 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ ≈ )
74, 6anbi12i 735 . . 3 ((𝐴𝐵 ∧ ¬ 𝐴𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ ≼ ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ ≈ ))
8 eldif 3725 . . 3 (⟨𝐴, 𝐵⟩ ∈ ( ≼ ∖ ≈ ) ↔ (⟨𝐴, 𝐵⟩ ∈ ≼ ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ ≈ ))
97, 8bitr4i 267 . 2 ((𝐴𝐵 ∧ ¬ 𝐴𝐵) ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≼ ∖ ≈ ))
102, 3, 93bitr4i 292 1 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 383  wcel 2139  cdif 3712  cop 4327   class class class wbr 4804  cen 8120  cdom 8121  csdm 8122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-dif 3718  df-br 4805  df-sdom 8126
This theorem is referenced by:  sdomdom  8151  sdomnen  8152  0sdomg  8256  sdomdomtr  8260  domsdomtr  8262  domtriord  8273  canth2  8280  php2  8312  php3  8313  nnsdomo  8322  nnsdomg  8386  card2inf  8627  cardsdomelir  9009  cardsdom2  9024  fidomtri2  9030  cardmin2  9034  alephordi  9107  alephord  9108  isfin4-3  9349  isfin5-2  9425  canthnum  9683  canthwe  9685  canthp1  9688  gchcdaidm  9702  gchxpidm  9703  gchhar  9713  axgroth6  9862  hashsdom  13382  ruc  15191
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