Theorem dfdom2 8974

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 8945	. . 3 ⊢ ≺ = ( ≼ ∖ ≈ )
2	1	uneq2i 4127	. 2 ⊢ ( ≈ ∪ ≺ ) = ( ≈ ∪ ( ≼ ∖ ≈ ))
3		uncom 4120	. 2 ⊢ ( ≈ ∪ ≺ ) = ( ≺ ∪ ≈ )
4		enssdom 8972	. . 3 ⊢ ≈ ⊆ ≼
5		undif 4448	. . 3 ⊢ ( ≈ ⊆ ≼ ↔ ( ≈ ∪ ( ≼ ∖ ≈ )) = ≼ )
6	4, 5	mpbi 233	. 2 ⊢ ( ≈ ∪ ( ≼ ∖ ≈ )) = ≼
7	2, 3, 6	3eqtr3ri 2801	1 ⊢ ≼ = ( ≺ ∪ ≈ )

Syntax hints:   = wceq 1567   ∖ cdif 3910   ∪ cun 3911   ⊆ wss 3913   ≈ 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-or 861  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-opab 5178  df-f1o 6544  df-en 8943  df-dom 8944  df-sdom 8945

This theorem is referenced by:  brdom2  8978