brsdom2 - Metamath Proof Explorer

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Theorem brsdom2 8684

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 8683	. . 3 ⊢ ≺ = ( ≼ ∖ ^◡ ≼ )
2	1	eleq2i 2824	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ ≺ ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≼ ∖ ^◡ ≼ ))
3		df-br 5028	. 2 ⊢ (𝐴 ≺ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≺ )
4		df-br 5028	. . . 4 ⊢ (𝐴 ≼ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≼ )
5		df-br 5028	. . . . . 6 ⊢ (𝐵 ≼ 𝐴 ↔ ⟨𝐵, 𝐴⟩ ∈ ≼ )
6		brsdom2.1	. . . . . . 7 ⊢ 𝐴 ∈ V
7		brsdom2.2	. . . . . . 7 ⊢ 𝐵 ∈ V
8	6, 7	opelcnv 5718	. . . . . 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 630	. . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐵 ≼ 𝐴) ↔ (⟨𝐴, 𝐵⟩ ∈ ≼ ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ ^◡ ≼ ))
12		eldif 3851	. . 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 399 ∈ wcel 2113 Vcvv 3397 ∖ cdif 3838 ⟨cop 4519 class class class wbr 5027 ^◡ccnv 5518 ≼ cdom 8546 ≺ csdm 8547

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551

This theorem is referenced by: (None)

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