brsdom2 - Metamath Proof Explorer

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Theorem brsdom2 9096

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 9095	. . 3 ⊢ ≺ = ( ≼ ∖ ^◡ ≼ )
2	1	eleq2i 2857	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ ≺ ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≼ ∖ ^◡ ≼ ))
3		df-br 5112	. 2 ⊢ (𝐴 ≺ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≺ )
4		df-br 5112	. . . 4 ⊢ (𝐴 ≼ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≼ )
5		df-br 5112	. . . . . 6 ⊢ (𝐵 ≼ 𝐴 ↔ ⟨𝐵, 𝐴⟩ ∈ ≼ )
6		brsdom2.1	. . . . . . 7 ⊢ 𝐴 ∈ V
7		brsdom2.2	. . . . . . 7 ⊢ 𝐵 ∈ V
8	6, 7	opelcnv 5869	. . . . . 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 640	. . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐵 ≼ 𝐴) ↔ (⟨𝐴, 𝐵⟩ ∈ ≼ ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ ^◡ ≼ ))
12		eldif 3916	. . 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

This proof depends on syntax axioms: ¬ wn 3 ↔ wb 209 ∧ wa 401 ∈ wcel 2146 Vcvv 3457 ∖ cdif 3903 ⟨cop 4597 class class class wbr 5111 ^◡ccnv 5662 ≼ cdom 8947 ≺ csdm 8948

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-pow 5338 ax-pr 5406 ax-un 7742

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ral 3082 df-rex 3092 df-rab 3419 df-v 3459 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-br 5112 df-opab 5176 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952

This theorem is used by: (None)

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