Theorem dfdom2 8139

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 8116	. . 3 ⊢ ≺ = ( ≼ ∖ ≈ )
2	1	uneq2i 3915	. 2 ⊢ ( ≈ ∪ ≺ ) = ( ≈ ∪ ( ≼ ∖ ≈ ))
3		uncom 3908	. 2 ⊢ ( ≈ ∪ ≺ ) = ( ≺ ∪ ≈ )
4		enssdom 8138	. . 3 ⊢ ≈ ⊆ ≼
5		undif 4192	. . 3 ⊢ ( ≈ ⊆ ≼ ↔ ( ≈ ∪ ( ≼ ∖ ≈ )) = ≼ )
6	4, 5	mpbi 220	. 2 ⊢ ( ≈ ∪ ( ≼ ∖ ≈ )) = ≼
7	2, 3, 6	3eqtr3ri 2802	1 ⊢ ≼ = ( ≺ ∪ ≈ )

Syntax hints:   = wceq 1631   ∖ cdif 3720   ∪ cun 3721   ⊆ wss 3723   ≈ cen 8110   ≼ cdom 8111   ≺ csdm 8112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-opab 4848  df-xp 5256  df-rel 5257  df-f1o 6037  df-en 8114  df-dom 8115  df-sdom 8116

This theorem is referenced by:  brdom2  8143