Theorem dfdom2 9016

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 4166	. 2 ⊢ ( ≈ ∪ ≺ ) = ( ≈ ∪ ( ≼ ∖ ≈ ))
3		uncom 4159	. 2 ⊢ ( ≈ ∪ ≺ ) = ( ≺ ∪ ≈ )
4		enssdom 9014	. . 3 ⊢ ≈ ⊆ ≼
5		undif 4440	. . 3 ⊢ ( ≈ ⊆ ≼ ↔ ( ≈ ∪ ( ≼ ∖ ≈ )) = ≼ )
6	4, 5	mpbi 233	. 2 ⊢ ( ≈ ∪ ( ≼ ∖ ≈ )) = ≼
7	2, 3, 6	3eqtr3ri 2795	1 ⊢ ≼ = ( ≺ ∪ ≈ )

This proof depends on syntax axioms:   = wceq 1569   ⊆ wss 3899   ∖ cdif 4047   ∪ cun 4151   ≈ cen 8981   ≼ cdom 8982   ≺ csdm 8983

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-opab 5170  df-f1o 6553  df-en 8985  df-dom 8986  df-sdom 8987

This theorem is used by:  brdom2  9020