Further Order Facts for ℕ₀ in Flat Form: ≤ as Membership or Equality, Transitivity for Sets, and the Order against +1

In the setting of Class Theory NBG: the Axioms, Standing Conventions and Basic Notation, let $\omega$ and $0$ be as in The Class Omega of Natural Numbers with Zero (i) and The Class Omega of Natural Numbers with Zero (ii), let $1$ be as in The Set of Natural Numbers and the Number One (ii), let $+$ be the addition on $\omega$, and let $\le$ and $<$ be the order and the strict order on $\omega$. Let $m,n\in\omega$ (so $1$, $m+1$ and $n+1$ lie in $\omega$ by Order and Arithmetic of ℕ₀ in Flat Form (ii)).

(i) $m\le n$ if and only if $m\in n$ or $m=n$.

(ii) For all sets $x$, $y$ and $z$: if $x\le y$ and $y\le z$, then $x\le z$.

(iii) $m+1\le n+1$ if and only if $m\le n$.

(iv) $m<n$ if and only if $m+1\le n$.

(v) $1\le n+1$.

(vi) If $n\neq0$, then $1\le n$.