brsdom2 - Metamath Proof Explorer

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Theorem brsdom2 9041

Description: Alternate definition of strict dominance. Definition 3 of [Suppes] p. 97. (Contributed by NM, 27-Jul-2004.)

**Hypotheses**
Ref	Expression
brsdom2.1	⊢ 𝐴 ∈ V
brsdom2.2	⊢ 𝐵 ∈ V

**Assertion**
Ref	Expression
brsdom2	⊢ (𝐴 ≺ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐵 ≼ 𝐴))

**Proof of Theorem brsdom2**
Step	Hyp	Ref	Expression
1		dfsdom2 9040	. . 3 ⊢ ≺ = ( ≼ ∖ ^◡ ≼ )
2	1	eleq2i 2829	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ ≺ ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≼ ∖ ^◡ ≼ ))
3		df-br 5101	. 2 ⊢ (𝐴 ≺ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≺ )
4		df-br 5101	. . . 4 ⊢ (𝐴 ≼ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≼ )
5		df-br 5101	. . . . . 6 ⊢ (𝐵 ≼ 𝐴 ↔ ⟨𝐵, 𝐴⟩ ∈ ≼ )
6		brsdom2.1	. . . . . . 7 ⊢ 𝐴 ∈ V
7		brsdom2.2	. . . . . . 7 ⊢ 𝐵 ∈ V
8	6, 7	opelcnv 5838	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ ^◡ ≼ ↔ ⟨𝐵, 𝐴⟩ ∈ ≼ )
9	5, 8	bitr4i 278	. . . . 5 ⊢ (𝐵 ≼ 𝐴 ↔ ⟨𝐴, 𝐵⟩ ∈ ^◡ ≼ )
10	9	notbii 320	. . . 4 ⊢ (¬ 𝐵 ≼ 𝐴 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ ^◡ ≼ )
11	4, 10	anbi12i 629	. . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐵 ≼ 𝐴) ↔ (⟨𝐴, 𝐵⟩ ∈ ≼ ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ ^◡ ≼ ))
12		eldif 3913	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ ( ≼ ∖ ^◡ ≼ ) ↔ (⟨𝐴, 𝐵⟩ ∈ ≼ ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ ^◡ ≼ ))
13	11, 12	bitr4i 278	. 2 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐵 ≼ 𝐴) ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≼ ∖ ^◡ ≼ ))
14	2, 3, 13	3bitr4i 303	1 ⊢ (𝐴 ≺ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐵 ≼ 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∈ wcel 2114 Vcvv 3442 ∖ cdif 3900 ⟨cop 4588 class class class wbr 5100 ^◡ccnv 5631 ≼ cdom 8893 ≺ csdm 8894

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-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-pow 5312 ax-pr 5379 ax-un 7690

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ral 3053 df-rex 3063 df-rab 3402 df-v 3444 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-id 5527 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898

This theorem is referenced by: (None)

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