Theorem dfdom2 9017

Description: Alternate definition of dominance. (Contributed by NM, 17-Jun-1998.)

Assertion

Ref	Expression
dfdom2	⊢ ≼ = ( ≺ ∪ ≈ )

Proof of Theorem dfdom2

Step	Hyp	Ref	Expression
1		df-sdom 8987	. . 3 ⊢ ≺ = ( ≼ ∖ ≈ )
2	1	uneq2i 4175	. 2 ⊢ ( ≈ ∪ ≺ ) = ( ≈ ∪ ( ≼ ∖ ≈ ))
3		uncom 4168	. 2 ⊢ ( ≈ ∪ ≺ ) = ( ≺ ∪ ≈ )
4		enssdom 9016	. . 3 ⊢ ≈ ⊆ ≼
5		undif 4488	. . 3 ⊢ ( ≈ ⊆ ≼ ↔ ( ≈ ∪ ( ≼ ∖ ≈ )) = ≼ )
6	4, 5	mpbi 230	. 2 ⊢ ( ≈ ∪ ( ≼ ∖ ≈ )) = ≼
7	2, 3, 6	3eqtr3ri 2772	1 ⊢ ≼ = ( ≺ ∪ ≈ )

Syntax hints:   = wceq 1537   ∖ cdif 3960   ∪ cun 3961   ⊆ wss 3963   ≈ cen 8981   ≼ cdom 8982   ≺ csdm 8983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211  df-xp 5695  df-rel 5696  df-f1o 6570  df-en 8985  df-dom 8986  df-sdom 8987

This theorem is referenced by:  brdom2  9021