# Metamath Proof Explorer

## Theorem entric

Description: Trichotomy of equinumerosity and strict dominance. This theorem is equivalent to the Axiom of Choice. Theorem 8 of Suppes p. 242. (Contributed by NM, 4-Jan-2004)

Ref Expression
Assertion entric ${⊢}\left({A}\in {V}\wedge {B}\in {W}\right)\to \left({A}\prec {B}\vee {A}\approx {B}\vee {B}\prec {A}\right)$

### Proof

Step Hyp Ref Expression
1 domtri ${⊢}\left({A}\in {V}\wedge {B}\in {W}\right)\to \left({A}\preccurlyeq {B}↔¬{B}\prec {A}\right)$
2 1 biimprd ${⊢}\left({A}\in {V}\wedge {B}\in {W}\right)\to \left(¬{B}\prec {A}\to {A}\preccurlyeq {B}\right)$
3 brdom2 ${⊢}{A}\preccurlyeq {B}↔\left({A}\prec {B}\vee {A}\approx {B}\right)$
4 2 3 syl6ib ${⊢}\left({A}\in {V}\wedge {B}\in {W}\right)\to \left(¬{B}\prec {A}\to \left({A}\prec {B}\vee {A}\approx {B}\right)\right)$
5 4 con1d ${⊢}\left({A}\in {V}\wedge {B}\in {W}\right)\to \left(¬\left({A}\prec {B}\vee {A}\approx {B}\right)\to {B}\prec {A}\right)$
6 5 orrd ${⊢}\left({A}\in {V}\wedge {B}\in {W}\right)\to \left(\left({A}\prec {B}\vee {A}\approx {B}\right)\vee {B}\prec {A}\right)$
7 df-3or ${⊢}\left({A}\prec {B}\vee {A}\approx {B}\vee {B}\prec {A}\right)↔\left(\left({A}\prec {B}\vee {A}\approx {B}\right)\vee {B}\prec {A}\right)$
8 6 7 sylibr ${⊢}\left({A}\in {V}\wedge {B}\in {W}\right)\to \left({A}\prec {B}\vee {A}\approx {B}\vee {B}\prec {A}\right)$