Theorem bren2 8912

Description: Equinumerosity expressed in terms of dominance and strict dominance. (Contributed by NM, 23-Oct-2004.)

Assertion

Ref	Expression
bren2	⊢ (𝐴 ≈ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≺ 𝐵))

Proof of Theorem bren2

Step	Hyp	Ref	Expression
1		endom 8908	. . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵)
2		sdomnen 8910	. . . 4 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵)
3	2	con2i 139	. . 3 ⊢ (𝐴 ≈ 𝐵 → ¬ 𝐴 ≺ 𝐵)
4	1, 3	jca 511	. 2 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≺ 𝐵))
5		brdom2 8911	. . . 4 ⊢ (𝐴 ≼ 𝐵 ↔ (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵))
6	5	biimpi 216	. . 3 ⊢ (𝐴 ≼ 𝐵 → (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵))
7	6	orcanai 1004	. 2 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≺ 𝐵) → 𝐴 ≈ 𝐵)
8	4, 7	impbii 209	1 ⊢ (𝐴 ≈ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≺ 𝐵))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∨ wo 847   class class class wbr 5093   ≈ cen 8872   ≼ cdom 8873   ≺ csdm 8874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-br 5094  df-opab 5156  df-f1o 6493  df-en 8876  df-dom 8877  df-sdom 8878

This theorem is referenced by:  marypha1lem  9324  tskwe  9850  infxpenlem  9911  cdainflem  10086  axcclem  10355  alephsuc3  10478  gchen1  10523  gchen2  10524  inatsk  10676  ufilen  23846  dirith2  27467  f1ocnt  32787  lindsenlbs  37675  mblfinlem1  37717  axccdom  45343  axccd2  45351