brsdom2 - Metamath Proof Explorer

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Theorem brsdom2 9085

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 9084	. . 3 ⊢ ≺ = ( ≼ ∖ ^◡ ≼ )
2	1	eleq2i 2855	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ ≺ ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≼ ∖ ^◡ ≼ ))
3		df-br 5110	. 2 ⊢ (𝐴 ≺ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≺ )
4		df-br 5110	. . . 4 ⊢ (𝐴 ≼ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≼ )
5		df-br 5110	. . . . . 6 ⊢ (𝐵 ≼ 𝐴 ↔ ⟨𝐵, 𝐴⟩ ∈ ≼ )
6		brsdom2.1	. . . . . . 7 ⊢ 𝐴 ∈ V
7		brsdom2.2	. . . . . . 7 ⊢ 𝐵 ∈ V
8	6, 7	opelcnv 5867	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ ^◡ ≼ ↔ ⟨𝐵, 𝐴⟩ ∈ ≼ )
9	5, 8	bitr4i 281	. . . . 5 ⊢ (𝐵 ≼ 𝐴 ↔ ⟨𝐴, 𝐵⟩ ∈ ^◡ ≼ )
10	9	notbii 323	. . . 4 ⊢ (¬ 𝐵 ≼ 𝐴 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ ^◡ ≼ )
11	4, 10	anbi12i 639	. . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐵 ≼ 𝐴) ↔ (⟨𝐴, 𝐵⟩ ∈ ≼ ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ ^◡ ≼ ))
12		eldif 3915	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ ( ≼ ∖ ^◡ ≼ ) ↔ (⟨𝐴, 𝐵⟩ ∈ ≼ ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ ^◡ ≼ ))
13	11, 12	bitr4i 281	. 2 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐵 ≼ 𝐴) ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≼ ∖ ^◡ ≼ ))
14	2, 3, 13	3bitr4i 306	1 ⊢ (𝐴 ≺ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐵 ≼ 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 209 ∧ wa 400 ∈ wcel 2143 Vcvv 3455 ∖ cdif 3902 ⟨cop 4595 class class class wbr 5109 ^◡ccnv 5660 ≼ cdom 8937 ≺ csdm 8938

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-pow 5336 ax-pr 5404 ax-un 7732

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-br 5110 df-opab 5174 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942

This theorem is referenced by: (None)

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