Theorem nehash2

Description: The cardinality of a set with two distinct elements. (Contributed by Thierry Arnoux, 27-Aug-2019)

	Ref	Expression
Hypotheses	nehash2.p	$$⊢\phi \to P\in V$$
	nehash2.a	$$⊢\phi \to A\in P$$
	nehash2.b	$$⊢\phi \to B\in P$$
	nehash2.1	$$⊢\phi \to A\ne B$$
Assertion	nehash2	$$⊢\phi \to 2\le \left|P\right|$$

Proof

Step	Hyp	Ref	Expression
1		nehash2.p	$$⊢\phi \to P\in V$$
2		nehash2.a	$$⊢\phi \to A\in P$$
3		nehash2.b	$$⊢\phi \to B\in P$$
4		nehash2.1	$$⊢\phi \to A\ne B$$
5		hashprg	$$⊢\left(A\in P\wedge B\in P\right)\to \left(A\ne B\leftrightarrow \left|\left\{A,B\right\}\right|=2\right)$$
6	2 3 5	syl2anc	$$⊢\phi \to \left(A\ne B\leftrightarrow \left|\left\{A,B\right\}\right|=2\right)$$
7	4 6	mpbid	$$⊢\phi \to \left|\left\{A,B\right\}\right|=2$$
8	2 3	prssd	$$⊢\phi \to \left\{A,B\right\}\subseteq P$$
9		hashss	$$⊢\left(P\in V\wedge \left\{A,B\right\}\subseteq P\right)\to \left|\left\{A,B\right\}\right|\le \left|P\right|$$
10	1 8 9	syl2anc	$$⊢\phi \to \left|\left\{A,B\right\}\right|\le \left|P\right|$$
11	7 10	eqbrtrrd	$$⊢\phi \to 2\le \left|P\right|$$