Theorem dfdom2 8919

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 8890	. . 3 ⊢ ≺ = ( ≼ ∖ ≈ )
2	1	uneq2i 4106	. 2 ⊢ ( ≈ ∪ ≺ ) = ( ≈ ∪ ( ≼ ∖ ≈ ))
3		uncom 4099	. 2 ⊢ ( ≈ ∪ ≺ ) = ( ≺ ∪ ≈ )
4		enssdom 8917	. . 3 ⊢ ≈ ⊆ ≼
5		undif 4423	. . 3 ⊢ ( ≈ ⊆ ≼ ↔ ( ≈ ∪ ( ≼ ∖ ≈ )) = ≼ )
6	4, 5	mpbi 230	. 2 ⊢ ( ≈ ∪ ( ≼ ∖ ≈ )) = ≼
7	2, 3, 6	3eqtr3ri 2769	1 ⊢ ≼ = ( ≺ ∪ ≈ )

Syntax hints:   = wceq 1542   ∖ cdif 3887   ∪ cun 3888   ⊆ wss 3890   ≈ cen 8884   ≼ cdom 8885   ≺ csdm 8886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-opab 5149  df-f1o 6500  df-en 8888  df-dom 8889  df-sdom 8890

This theorem is referenced by:  brdom2  8923